Internet-Draft | VDAF | October 2024 |
Barnes, et al. | Expires 7 April 2025 | [Page] |
This document describes Verifiable Distributed Aggregation Functions (VDAFs), a family of multi-party protocols for computing aggregate statistics over user measurements. These protocols are designed to ensure that, as long as at least one aggregation server executes the protocol honestly, individual measurements are never seen by any server in the clear. At the same time, VDAFs allow the servers to detect if a malicious or misconfigured client submitted an measurement that would result in an invalid aggregate result.¶
This note is to be removed before publishing as an RFC.¶
Discussion of this document takes place on the Crypto Forum Research Group mailing list (cfrg@ietf.org), which is archived at https://mailarchive.ietf.org/arch/search/?email_list=cfrg.¶
Source for this draft and an issue tracker can be found at https://github.com/cjpatton/vdaf.¶
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Copyright (c) 2024 IETF Trust and the persons identified as the document authors. All rights reserved.¶
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(RFC EDITOR: remove this paragraph.) The source for this draft and and the reference implementation can be found at https://github.com/cfrg/draft-irtf-cfrg-vdaf.¶
The ubiquity of the Internet makes it an ideal platform for measurement of large-scale phenomena, whether public health trends or the behavior of computer systems at scale. There is substantial overlap, however, between information that is valuable to measure and information that users consider private.¶
For example, consider an application that provides health information to users. The operator of an application might want to know which parts of their application are used most often, as a way to guide future development of the application. Specific users' patterns of usage, though, could reveal sensitive things about them, such as which users are researching a given health condition.¶
In many situations, the measurement collector is only interested in aggregate statistics, e.g., which portions of an application are most used or what fraction of people have experienced a given disease. Thus systems that provide aggregate statistics while protecting individual measurements can deliver the value of the measurements while protecting users' privacy.¶
This problem is often formulated in terms of differential privacy (DP) [Dwo06]. Roughly speaking, a data aggregation system that is differentially private ensures that the degree to which any individual measurement influences the value of the aggregate result can be precisely controlled. For example, in systems like RAPPOR [EPK14], each user samples noise from a well-known distribution and adds it to their measurement before submitting to the aggregation server. The aggregation server then adds up the noisy measurements, and because it knows the distribution from which the noise was sampled, it can estimate the true sum with reasonable accuracy.¶
Even when noise is added to the measurements, collecting them in the clear still reveals a significant amount of information to the collector. On the one hand, depending on the "amount" of noise a client adds to its measurement, it may be possible for a curious collector to make a reasonable guess of the measurement's true value. On the other hand, the more noise the clients add, the less reliable will be the server's estimate of the aggregate. Thus systems relying solely on a DP mechanism must strike a delicate balance between privacy and utility.¶
The ideal goal for a privacy-preserving measurement system is that of secure multi-party computation (MPC): no participant in the protocol should learn anything about an individual measurement beyond what it can deduce from the aggregate. MPC achieves this goal by distributing the computation of the aggregate across multiple aggregation servers, one of which is presumed to be honest, i.e., not under control of the attacker. Moreover, MPC can be composed with various DP mechanisms to ensure the aggregate itself does leak too much information about any one of the measurements [MPRV09].¶
This document describes two classes of MPC protocols, each aiming for a different set of goals.¶
In a Distributed Aggregation Function (DAF, Section 4), each client splits its measurement into multiple secret shares, one for each aggregation server. DAFs require two properties of the secret sharing scheme. First, we can reconstruct the underlying measurement by simply adding up all of the shares. (Typically the shares are vectors over some finite field.) Second, given all but one of the shares, it is impossible to learn anything about the underlying measurement. These properties give rise to a simple strategy for privately aggregating the measurements: each aggregation server adds up its measurement shares locally before revealing their sum to the data collector; then all the data collector has to do is add up these sums to get the aggregate.¶
This strategy is compatible with any aggregation function that can be represented as the sum of some encoding of the measurements. Examples include: summary statistics such as sum, mean, and standard deviation; estimation of quantiles, e.g., median; histograms; linear regression; or counting data structures, e.g., Bloom filters. However, not all functions fit into this rubric, as it is constrained to linear computations over the encoded measurements.¶
In fact, our framework admits DAFs with slightly more functionality, computing aggregation functions of the form¶
f(agg_param, meas_1, ..., meas_N) = g(agg_param, meas_1) + ... + g(agg_param, meas_N)¶
where meas_1, ..., meas_N
are the measurements, g
is a possibly non-linear
function, and agg_param
is a parameter of that function chosen by the data
collector. This paradigm, known as function secret sharing [BGI15], allows
for more sophisticated data analysis tasks, such as grouping metrics by private
client attributes [MPDST25] or computing heavy hitters [BBCGGI21]. (More
on the latter task below.)¶
The second class of protocols defined in this document are called Verifiable Distributed Aggregation Functions (VDAFs, Section 5). In addition to being private, VDAFs are verifiable in the following sense. By design, a secret sharing of a valid measurement, e.g., a number between 1 and 10, is indistinguishable from a secret sharing of an invalid measurement, e.g., a number larger than 10. This means that DAFs are vulnerable to attacks from malicious clients attempting to disrupt the computation by submitting invalid measurements. Thus VDAFs are designed to allow the servers to detect and remove these measurements prior to aggregation. We refer to this property as robustness.¶
Achieving robustness without sacrificing privacy requires the servers to interact with one another over a number of rounds of communication. DAFs on the other hand are non-interactive, and are therefore easier to deploy; but they do not provide robustness on their own. This may be tolerable in some applications. For instance, if the client's software is executed in a trusted execution environment, it may be reasonable to assume that no client is malicious.¶
The DAF and VDAF abstractions encompass a variety of MPC techniques in the literature. These protocols vary in their operational and security requirements, sometimes in subtle but consequential ways. This document therefore has two important goals:¶
Providing higher-level protocols like [DAP] (RFC EDITOR: remove this reference if not published before the current document) with a simple, uniform interface for accessing privacy-preserving measurement schemes, documenting relevant operational and security requirements, and specifying constraints for safe usage:¶
General patterns of communications among the various actors involved in the system (clients, aggregation servers, and the collector of the aggregate result);¶
Capabilities of a malicious coalition of servers attempting to divulge information about client measurements; and¶
Conditions that are necessary to ensure that malicious clients cannot corrupt the computation.¶
Providing cryptographers with design criteria that provide a clear deployment roadmap for new constructions.¶
This document also specifies two concrete VDAF schemes, each based on a protocol from the literature.¶
The Prio system [CGB17] allows for the privacy-preserving computation of a variety of aggregate statistics, combining additive secret sharing as described above with a mechanism for checking the validity of each measurement. In Section 7 we specify Prio3, a VDAF that follows the same overall framework as the original Prio protocol, but incorporates techniques introduced in [BBCGGI19] that result in significant performance gains.¶
The Poplar protocol [BBCGGI21] solves the heavy-hitters problem in a
privacy-preserving manner. Here each client holds a bit-string, and the goal
of the aggregation servers is to compute the set of strings that occur at
least t
times for some threshold t
. The core primitive in their protocol
is a secret sharing of a point function [GI14] (g
in the notation above)
that allows the servers to privately count how many of the clients' strings
begin with a given prefix (agg_param
in the notation above). In Section 8
we specify a VDAF called Poplar1 that implements this functionality.¶
The remainder of this document is organized as follows: Section 2 lists definitions and conventions used for specification; Section 3 gives a brief overview of DAFs and VDAFs, the parties involved in the computation, and the requirements for non-collusion; Section 4 defines the syntax for DAFs; Section 5 defines the syntax for VDAFs; Section 6 defines various functionalities that are common to our constructions; Section 7 describes the Prio3 construction; Section 8 describes the Poplar1 construction; and Section 9 enumerates the security considerations for DAFs and VDAFs.¶
(RFC EDITOR: remove this section.)¶
(*) Indicates a change that breaks wire compatibility with the previous draft.¶
12:¶
(V)DAF: Add an application context string parameter to sharding and preparation. The motivation for this change is to harden Prio3 against offline attacks. More generally, however, it allows designing schemes for which correct execution requires agreement on the application context. Accordingly, both Prio3 and Poplar1 have been modified to include the context in the domain separation tag of each XOF invocation. (*)¶
Prio3: Improve soundness of the base proof system and the circuits of some variants. Generally speaking, wherever we evaluate a univariate polynomial at a random point, we can instead evaluate a multivariate polynomial of lower degree. (*)¶
Prio3: Replace the helper's measurement and proof share seeds with a single seed. (*)¶
Prio3Sum: Update the circuit to support a more general range check and avoid using joint randomness. (*)¶
Prio3Histogram, Prio3MultihotCountVec: Move the final reduction of the intermediate outputs out of the circuit. (*)¶
IDPF: Add the application context string to key generation end evaluation and bind it to the fixed AES key. (*)¶
IDPF: Use XofTurboShake128 for deriving the leaf nodes in order to ensure the construction is extractable. (*)¶
IDPF: Simplify the public share encoding. (*)¶
XofTurboShake128: Change SEED_SIZE
from 16 bytes to 32 to mitigate offline
attacks on Prio3 robustness. In addition, allow seeds of different lengths so
that we can continue to use XofTurboShake128 with IDPF. (*)¶
XofTurboShake128, XofFixedKeyAes128: Increase the length prefix for the domain separation tag from one by to two bytes. This is to accommodate the application context. (*)¶
Reassign codepoints for all Prio3 variants and Poplar1. (*)¶
Security considerations: Add a section on defense-in-depth measures taken by Prio3 and Poplar1 and more discussion about choosing FLP parameters.¶
11:¶
Define message formats for the Poplar1 aggregation parameter and IDPF public share.¶
IDPF: Require the IDPF binder must be a random nonce.¶
VDAF: Replace the pseudocode description of the ping-ping topology with Python and sketch the star topology.¶
DAF: Align aggregation parameter validation with VDAF.¶
Replace Union[A, B]
type with A | B
.¶
Rename FFT ("Fast Fourier Transform") with NTT ("Number Theoretic Transform").¶
10:¶
Define Prio3MultihotCountVec, a variant of Prio3 for aggregating bit vectors with bounded weight.¶
FLP: Allow the output of the circuit to be a vector. This makes it possible to skip joint randomness derivation in more cases.¶
Poplar1: On the first round of preparation, handle None
as an error.
Previously this message was interpreted as a length-3 vector of zeros.¶
Prio3: Move specification of the field from the FLP validity circuit to the VDAF itself.¶
Clarify the extent to which the attacker controls the network in our threat models for privacy and robustness.¶
Clean up various aspects of the code, including: Follow existing object-oriented programming patterns for Python more closely; make the type hints enforceable; and avoid shadowing variables.¶
IDPF: Add guidance for encoding byte strings as indices.¶
09:¶
Poplar1: Make prefix tree traversal stricter by requiring each node to be a child of a node that was already visited. This change is intended to make it harder for a malicious Aggregator to steer traversal towards non-heavy-hitting measurements.¶
Prio3: Add more explicit guidance for choosing the field size.¶
IDPF: Define extractability and clarify (un)safe usage of intermediate prefix counts. Accordingly, add text ensuring public share consistency to security considerations.¶
08:¶
Poplar1: Bind the report nonce to the authenticator vector programmed into the IDPF. (*)¶
IdpfPoplar: Modify extend()
by stealing each control bit from its
corresponding seed. This improves performance by reducing the number of AES
calls per level from 3 to 2. The cost is a slight reduction in the concrete
privacy bound. (*)¶
Prio3: Add support for generating and verifying mutliple proofs per measurement. This enables a trade-off between communication cost and runtime: if more proofs are used, then a smaller field can be used without impacting robustness. (*)¶
Replace SHAKE128 with TurboSHAKE128. (*)¶
07:¶
Rename PRG to XOF ("eXtendable Output Function"). Accordingly, rename PrgSha3 to XofShake128 and PrgFixedKeyAes128 to XofFixedKeyAes128. "PRG" is a misnomer since we don't actually treat this object as a pseudorandom generator in existing security analysis.¶
Replace cSHAKE128 with SHAKE128, re-implementing domain separation for the customization string using a simpler scheme. This change addresses the reality that implementations of cSHAKE128 are less common. (*)¶
Define a new VDAF, called Prio3SumVec, that generalizes Prio3Sum to a vector of summands.¶
Prio3Histogram: Update the codepoint and use the parallel sum optimization introduced by Prio3SumVec to reduce the proof size. (*)¶
Daf, Vdaf: Rename interface methods to match verbiage in the draft.¶
Daf: Align with Vdaf by adding a nonce to shard()
and prep()
.¶
Vdaf: Have prep_init()
compute the first prep share. This change is
intended to simplify the interface by making the input to prep_next()
not
optional.¶
Prio3: Split sharding into two auxiliary functions, one for sharding with joint randomness and another without. This change is intended to improve readability.¶
Fix bugs in the ping-pong interface discovered after implementing it.¶
06:¶
Vdaf: Define a wrapper interface for preparation that is suitable for the "ping-pong" topology in which two Aggregators exchange messages over a request/response protocol, like HTTP, and take turns executing the computation until input from the peer is required.¶
Prio3Histogram: Generalize the measurement type so that the histogram can be used more easily with discrete domains. (*)¶
Daf, Vdaf: Change the aggregation parameter validation algorithm to take the set of previous parameters rather than a list. (The order of the parameters is irrelevant.)¶
Daf, Vdaf, Idpf: Add parameter RAND_SIZE
that specifies the number of
random bytes consumed by the randomized algorithm (shard()
for Daf and Vdaf
and gen()
for Idpf).¶
05:¶
IdpfPoplar: Replace PrgSha3 with PrgFixedKeyAes128, a fixed-key mode for AES-128 based on a construction from [GKWWY20]. This change is intended to improve performance of IDPF evaluation. Note that the new PRG is not suitable for all applications. (*)¶
Idpf: Add a binder string to the key-generation and evaluation algorithms. This is used to plumb the nonce generated by the Client to the PRG.¶
Plumb random coins through the interface of randomized algorithms. Specifically, add a random input to (V)DAF sharding algorithm and IDPF key-generation algorithm and require implementations to specify the length of the random input. Accordingly, update Prio3, Poplar1, and IdpfPoplar to match the new interface. This change is intended to improve coverage of test vectors.¶
Use little-endian byte-order for field element encoding. (*)¶
Poplar1: Move the last step of sketch evaluation from prep_next()
to
prep_shares_to_prep()
.¶
04:¶
Align security considerations with the security analysis of [DPRS23].¶
Vdaf: Pass the nonce to the sharding algorithm.¶
Vdaf: Rather than allow the application to choose the nonce length, have each implementation of the Vdaf interface specify the expected nonce length. (*)¶
Prg: Split "info string" into two components: the "customization string", intended for domain separation; and the "binder string", used to bind the output to ephemeral values, like the nonce, associated with execution of a (V)DAF.¶
Replace PrgAes128 with PrgSha3, an implementation of the Prg interface based on SHA-3, and use the new scheme as the default. Accordingly, replace Prio3Aes128Count with Prio3Count, Poplar1Aes128 with Poplar1, and so on. SHA-3 is a safer choice for instantiating a random oracle, which is used in the analysis of Prio3 of [DPRS23]. (*)¶
Prio3, Poplar1: Ensure each invocation of the Prg uses a distinct customization string, as suggested by [DPRS23]. This is intended to make domain separation clearer, thereby simplifying security analysis. (*)¶
Prio3: Replace "joint randomness hints" sent in each input share with "joint randomness parts" sent in the public share. This reduces communication overhead when the number of shares exceeds two. (*)¶
Prio3: Bind nonce to joint randomness parts. This is intended to address birthday attacks on robustness pointed out by [DPRS23]. (*)¶
Poplar1: Use different Prg invocations for producing the correlated randomness for inner and leaf nodes of the IDPF tree. This is intended to simplify implementations. (*)¶
Poplar1: Don't bind the candidate prefixes to the verifier randomness. This is intended to improve performance, while not impacting security. According to the analysis of [DPRS23], it is necessary to restrict Poplar1 usage such that no report is aggregated more than once at a given level of the IDPF tree; otherwise, attacks on privacy may be possible. In light of this restriction, there is no added benefit of binding to the prefixes themselves. (*)¶
Poplar1: During preparation, assert that all candidate prefixes are unique and appear in order. Uniqueness is required to avoid erroneously rejecting a valid report; the ordering constraint ensures the uniqueness check can be performed efficiently. (*)¶
Poplar1: Increase the maximum candidate prefix count in the encoding of the aggregation parameter. (*)¶
Poplar1: Bind the nonce to the correlated randomness derivation. This is intended to provide defense-in-depth by ensuring the Aggregators reject the report if the nonce does not match what the Client used for sharding. (*)¶
Poplar1: Clarify that the aggregation parameter encoding is OPTIONAL. Accordingly, update implementation considerations around cross-aggregation state.¶
IdpfPoplar: Add implementation considerations around branching on the values of control bits.¶
IdpfPoplar: When decoding the the control bits in the public share, assert that the trailing bits of the final byte are all zero. (*)¶
03:¶
Define codepoints for (V)DAFs and use them for domain separation in Prio3 and Poplar1. (*)¶
Prio3: Align joint randomness computation with revised paper [BBCGGI19]. This change mitigates an attack on robustness. (*)¶
Prio3: Remove an intermediate PRG evaluation from query randomness generation. (*)¶
Add additional guidance for choosing FFT-friendly fields.¶
02:¶
Complete the initial specification of Poplar1.¶
Extend (V)DAF syntax to include a "public share" output by the Client and distributed to all of the Aggregators. This is to accommodate "extractable" IDPFs as required for Poplar1. (See [BBCGGI21], Section 4.3 for details.)¶
Extend (V)DAF syntax to allow the unsharding step to take into account the number of measurements aggregated.¶
Extend FLP syntax by adding a method for decoding the aggregate result from a vector of field elements. The new method takes into account the number of measurements.¶
Prio3: Align aggregate result computation with updated FLP syntax.¶
Prg: Add a method for statefully generating a vector of field elements.¶
Field: Require that field elements are fully reduced before decoding. (*)¶
Define new field Field255.¶
01:¶
Require that VDAFs specify serialization of aggregate shares.¶
Define Distributed Aggregation Functions (DAFs).¶
Prio3: Move proof verifier check from prep_next()
to
prep_shares_to_prep()
. (*)¶
Remove public parameter and replace verification parameter with a "verification key" and "Aggregator ID".¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
Algorithms in this document are written in Python 3. Type hints are used to
define input and output types. The type variable F
is used in signatures to
signify any type that is a subclass of Field
. A fatal error in a program
(e.g., failure to parse one of the function parameters) is usually handled by
raising an exception.¶
A variable with type bytes
is a byte string. This document defines several
byte-string constants. When comprised of printable ASCII characters, they are
written as Python 3 byte-string literals (e.g., b'some constant string'
).¶
A global constant VERSION
of type int
is defined, which algorithms are
free to use as desired. Its value SHALL be 12
.¶
This document describes algorithms for multi-party computations in which the parties typically communicate over a network. Wherever a quantity is defined that must be be transmitted from one party to another, this document prescribes a particular encoding of that quantity as a byte string.¶
OPEN ISSUE It might be better to not be prescriptive about how quantities are encoded on the wire. See issue #58.¶
Some common functionalities:¶
zeros(len: int) -> bytes
returns an array of zero bytes. The length of
output
MUST be len
.¶
gen_rand(len: int) -> bytes
returns an array of random bytes generated by a
cryptographically secure pseudorandom number generator (CSPRNG). The length
of output
MUST be len
.¶
byte(int: int) -> bytes
returns the representation of int
as a byte
string. The value of int
MUST be in [0,256)
.¶
concat(parts: list[bytes]) -> bytes
returns the concatenation of the input
byte strings, i.e., parts[0] || ... || parts[len(parts)-1]
.¶
front(length: int, vec: list[Any]) -> (list[Any], list[Any])
splits vec
into two vectors, where the first vector is made up of the first length
elements of the input. I.e., (vec[:length], vec[length:])
.¶
xor(left: bytes, right: bytes) -> bytes
returns the bitwise XOR of left
and right
. An exception is raised if the inputs are not the same length.¶
to_be_bytes(val: int, length: int) -> bytes
converts val
to big-endian
bytes; its value MUST be at least 0
but less than 2**(8*length)
. Function
from_be_bytes(encoded: bytes) -> int
computes the inverse.¶
to_le_bytes(val: int, length: int) -> bytes
converts val
to little-endian
bytes; its value MUST be at least 0
but less than 2**(8*length)
. Function
from_le_bytes(encoded: bytes) -> int
computes the inverse.¶
next_power_of_2(n: int) -> int
returns the smallest integer
greater than or equal to n
that is also a power of two.¶
additive_secret_share(vec: list[Field], num_shares: int, field: type)
-> list[list[Field]]
takes a vector of field elements and returns multiple
vectors of the same length, such that they all add up to the input vector,
and each proper subset of the vectors are indistinguishable from random.¶
cast(typ: type, val: object) -> object
returns the input value unchanged.
This is only present to assist with static analysis of the Python code.
Type checkers will ignore the inferred type of the input value, and assume
the output value has the given type.¶
range(stop)
or range(start, stop[, step])
is the range function from the
Python standard library. The one-argument form returns the integers from zero
(inclusive) to stop
, exclusive. The two- and three-argument forms allow
overriding the start of the range and overriding the step between successive
output values.¶
In a DAF- or VDAF-based private measurement system, we distinguish three types of actors: Clients, Aggregators, and Collectors. The overall flow of the measurement process is as follows:¶
To submit an individual measurement, the Client shards the measurement into "input shares" and sends one input share to each Aggregator. We sometimes refer to this sequence of input shares collectively as the Client's "report".¶
The Aggregators refine their input shares into "output shares".¶
Output shares are in one-to-one correspondence with the input shares.¶
Just as each Aggregator receives one input share of each measurement, if this process succeeds, then each aggregator holds one output share.¶
In VDAFs, Aggregators will need to exchange information among themselves as part of the validation process.¶
Each Aggregator combines the output shares in the batch to compute the "aggregate share" for that batch, i.e., its share of the desired aggregate result.¶
The Aggregators submit their aggregate shares to the Collector, who combines them to obtain the aggregate result over the batch.¶
Aggregators are a new class of actor relative to traditional measurement systems where Clients submit measurements to a single server. They are critical for both the privacy properties of the system and, in the case of VDAFs, the correctness of the measurements obtained. The privacy properties of the system are assured by non-collusion among Aggregators, and Aggregators are the entities that perform validation of Client measurements. Thus Clients trust Aggregators not to collude (typically it is required that at least one Aggregator is honest; see Section 9.7), and Collectors trust Aggregators to correctly run the protocol.¶
Within the bounds of the non-collusion requirements of a given (V)DAF instance, it is possible for the same entity to play more than one role. For example, the Collector could also act as an Aggregator, effectively using the other Aggregator(s) to augment a basic client-server protocol.¶
In this document, we describe the computations performed by the actors in this system. It is up to the higher-level protocol making use of the (V)DAF to arrange for the required information to be delivered to the proper actors in the proper sequence. In general, we assume that all communications are confidential and mutually authenticated, with the exception that Clients submitting measurements may be anonymous.¶
By way of a gentle introduction to VDAFs, this section describes a simpler class of schemes called Distributed Aggregation Functions (DAFs). Unlike VDAFs, DAFs do not provide verifiability of the computation. Clients must therefore be trusted to compute their input shares correctly. Because of this fact, the use of a DAF is NOT RECOMMENDED for most applications. See Section 9 for additional discussion.¶
A DAF scheme is used to compute a particular "aggregation function" over a set of measurements generated by Clients. Depending on the aggregation function, the Collector might select an "aggregation parameter" and disseminates it to the Aggregators. The semantics of this parameter is specific to the aggregation function, but in general it is used to represent the set of "queries" that can be made on the measurement set. For example, the aggregation parameter is used to represent the candidate prefixes in Poplar1 Section 8.¶
Execution of a DAF has four distinct stages:¶
Sharding - Each Client generates input shares from its measurement and distributes them among the Aggregators.¶
Preparation - Each Aggregator converts each input share into an output share compatible with the aggregation function. This computation involves the aggregation parameter. In general, each aggregation parameter may result in a different an output share.¶
Aggregation - Each Aggregator combines a sequence of output shares into its aggregate share and sends the aggregate share to the Collector.¶
Unsharding - The Collector combines the aggregate shares into the aggregate result.¶
Sharding and Preparation are done once per measurement. Aggregation and Unsharding are done over a batch of measurements (more precisely, over the recovered output shares).¶
A concrete DAF specifies an algorithm for the computation needed in each of these stages. The interface of each algorithm is defined in the remainder of this section. In addition, a concrete DAF defines the associated constants and types enumerated in the following table.¶
Parameter | Description |
---|---|
ID: int
|
Algorithm identifier for this DAF, in range(2**32) . |
SHARES: int
|
Number of input shares into which each measurement is sharded. |
NONCE_SIZE: int
|
Size of the nonce passed by the application. |
RAND_SIZE: int
|
Size of the random byte string passed to sharding algorithm. |
Measurement
|
Type of each measurement. |
PublicShare
|
Type of each public share. |
InputShare
|
Type of each input share. |
AggParam
|
Type of aggregation parameter. |
OutShare
|
Type of each output share. |
AggShare
|
Type of the aggregate share. |
AggResult
|
Type of the aggregate result. |
These types define the inputs and outputs of DAF methods at various stages of
the computation. Some of these values need to be written to the network in
order to carry out the computation. In particular, it is RECOMMENDED that
concrete instantiations of the Daf
interface specify a method of encoding the
PublicShare
, InputShare
, and AggShare
.¶
Each DAF is identified by a unique, 32-bit integer ID
. Identifiers for each
(V)DAF specified in this document are defined in Table 23.¶
In order to protect the privacy of its measurements, a DAF Client shards its
measurements into a sequence of input shares. The shard
method is used for
this purpose.¶
daf.shard(ctx: bytes, measurement: Measurement, nonce: bytes, rand: bytes)
-> tuple[PublicShare, list[InputShare]]
is the randomized sharding algorithm
run by each Client that consumes the application context, a measurement, and
a nonce and produces a "public share" distributed to each of the Aggregate
and a corresponding sequence of input shares, one for each Aggregator.¶
Pre-conditions:¶
nonce
MUST have length equal to NONCE_SIZE
and MUST be generated using
a CSPRNG.¶
rand
consists of the random bytes consumed by the algorithm. It MUST have
length equal to RAND_SIZE
and MUST be generated using a CSPRNG.¶
Post-conditions:¶
The number of input shares MUST equal SHARES
.¶
Sharding is bound to a specific "application context". The application context is a string intended to uniquely identify an instance of the higher level protocol that uses the DAF. This is intended to ensure that aggregation succeeds only if the Clients and Aggregators agree on the application context. (Preparation binds the application context, too; see Section 4.2.) Note that, unlike VDAFs (Section 5), there is no explicit signal of disagreement; it may only manifest as a garbled aggregate result.¶
Once an Aggregator has received the public share and one of the input shares, the next step is to prepare the input share for aggregation. This is accomplished using the following algorithm:¶
daf.prep(ctx: bytes, agg_id: int, agg_param: AggParam, nonce: bytes,
public_share: PublicShare, input_share: InputShare) -> OutShare
is the
deterministic preparation algorithm. It takes as input the public share and
one of the input shares generated by a Client, the application context, the
Aggregator's unique identifier, the aggregation parameter selected by the
Collector, and a nonce and returns an output share.¶
Pre-conditions:¶
Concrete DAFs implementations MAY impose certain restrictions for input shares
and aggregation parameters. Protocols using a DAF MUST ensure that for each
input share and aggregation parameter agg_param
, daf.prep
is only called if
daf.is_valid(agg_param, previous_agg_params)
returns True, where
previous_agg_params
contains all aggregation parameters that have previously
been used with the same input share.¶
DAFs MUST implement the following function:¶
daf.is_valid(agg_param: AggParam, previous_agg_params: list[AggParam]) ->
bool
: checks if the agg_param
is compatible with all elements of
previous_agg_params
.¶
Once an Aggregator holds output shares for a batch of measurements (where batches are defined by the application), it combines them into a share of the desired aggregate result:¶
daf.aggregate(agg_param: AggParam, out_shares: list[OutShare]) -> AggShare
is the deterministic aggregation algorithm. It is run by each Aggregator a
set of recovered output shares.¶
For simplicity, we have written this algorithm in a "one-shot" form, where all output shares for a batch are provided at the same time. Many DAFs may also support a "streaming" form, where shares are processed one at a time.¶
Implementation note: for most natural DAFs (and VDAFs) it is not necessary for an Aggregator to store all output shares individually before aggregating. Typically it is possible to merge output shares into aggregate shares as they arrive, merge these into other aggregate shares, and so on. In particular, this is the case when the output shares are vectors over some finite field and aggregating them involves merely adding up the vectors element-wise. Such is the case for Prio3 Section 7 and Poplar1 Section 8.¶
After the Aggregators have aggregated a sufficient number of output shares, each sends its aggregate share to the Collector, who runs the following algorithm to recover the following output:¶
daf.unshard(agg_param: AggParam, agg_shares: list[AggShare],
num_measurements: int) -> AggResult
is run by the Collector in order to
compute the aggregate result from the Aggregators' shares. The length of
agg_shares
MUST be SHARES
. num_measurements
is the number of
measurements that contributed to each of the aggregate shares. This algorithm
is deterministic.¶
Securely executing a DAF involves emulating the following procedure.¶
The inputs to this procedure are the same as the aggregation function computed by the DAF: an aggregation parameter and a sequence of measurements. The procedure prescribes how a DAF is executed in a "benign" environment in which there is no adversary and the messages are passed among the protocol participants over secure point-to-point channels. In reality, these channels need to be instantiated by some "wrapper protocol", such as [DAP], that realizes these channels using suitable cryptographic mechanisms. Moreover, some fraction of the Aggregators (or Clients) may be malicious and diverge from their prescribed behaviors. Section 9 describes the execution of the DAF in various adversarial environments and what properties the wrapper protocol needs to provide in each.¶
Like DAFs described in the previous section, a VDAF scheme is used to compute a particular aggregation function over a set of Client-generated measurements. Evaluation of a VDAF involves the same four stages as for DAFs: Sharding, Preparation, Aggregation, and Unsharding. However, the Preparation stage will require interaction among the Aggregators in order to facilitate verifiability of the computation's correctness. Accommodating this interaction will require syntactic changes.¶
Overall execution of a VDAF comprises the following stages:¶
Sharding - Computing input shares from an individual measurement¶
Preparation - Conversion and verification of input shares to output shares compatible with the aggregation function being computed¶
Aggregation - Combining a sequence of output shares into an aggregate share¶
Unsharding - Combining a sequence of aggregate shares into an aggregate result¶
In contrast to DAFs, the Preparation stage for VDAFs now performs an additional task: verification of the validity of the recovered output shares. This process ensures that aggregating the output shares will not lead to a garbled aggregate result.¶
The remainder of this section defines the VDAF interface. The attributes are listed in Table 2 are defined by each concrete VDAF.¶
Parameter | Description |
---|---|
ID
|
Algorithm identifier for this VDAF. |
VERIFY_KEY_SIZE
|
Size (in bytes) of the verification key (Section 5.2). |
RAND_SIZE
|
Size of the random byte string passed to sharding algorithm. |
NONCE_SIZE
|
Size (in bytes) of the nonce. |
ROUNDS
|
Number of rounds of communication during the Preparation stage (Section 5.2). |
SHARES
|
Number of input shares into which each measurement is sharded (Section 5.1). |
Measurement
|
Type of each measurement. |
PublicShare
|
Type of each public share. |
InputShare
|
Type of each input share. |
AggParam
|
Type of aggregation parameter. |
OutShare
|
Type of each output share. |
AggShare
|
Type of the aggregate share. |
AggResult
|
Type of the aggregate result. |
PrepState
|
Aggregator's state during preparation. |
PrepShare
|
Type of each prep share. |
PrepMessage
|
Type of each prep message. |
Some of these values need to be written to the network in order to carry out
the computation. In particular, it is RECOMMENDED that concrete instantiations
of the Vdaf
interface specify a method of encoding the PublicShare
,
InputShare
, AggShare
, PrepShare
, and PrepMessage
.¶
Each VDAF is identified by a unique, 32-bit integer ID
. Identifiers for each
(V)DAF specified in this document are defined in Table 23. The following
method is defined for each VDAF specified in this document:¶
def domain_separation_tag(self, usage: int, ctx: bytes) -> bytes: """ Format domain separation tag for this VDAF with the given application context and usage. Pre-conditions: - `usage` in `range(2**16)` """ return format_dst(0, self.ID, usage) + ctx¶
It is used to construct a domain separation tag for an instance of Xof
used by
the VDAF. (See Section 6.2.)¶
Sharding transforms a measurement and nonce into a public share and input shares as it does in DAFs (cf. Section 4.1):¶
vdaf.shard(ctx: bytes, measurement: Measurement, nonce: bytes, rand: bytes)
-> tuple[PublicShare, list[InputShare]]
is the randomized sharding algorithm
run by each Client that consumes the application context, a measurement, and
a nonce and produces a public share distributed to each of the Aggregate and
a corresponding sequence of input shares, one for each Aggregator. Depending
on the VDAF, the input shares may encode additional information used to
verify the recovered output shares (e.g., the "proof shares" in Prio3
Section 7)¶
Pre-conditions:¶
nonce
MUST have length equal to NONCE_SIZE
and MUST be generated using
a CSPRNG. (See Section 9 for details.)¶
rand
consists of the random bytes consumed by the algorithm. It MUST have
length equal to RAND_SIZE
and MUST be generated using a CSPRNG.¶
Post-conditions:¶
The number of input shares MUST equal SHARES
.¶
Like DAFs, sharding is bound to the application context via the ctx
string.
Again, this is intended to ensure that aggregation succeeds only if the Clients
and Aggregators agree on the application context. Unlike DAFs, however,
disagreement on the context should manifest as a failure to validate the
report, causing the report to be rejected without garbling the aggregate
result. The application context also provides some defense-in-depth against
cross protocol attacks; see Section 9.8.¶
To recover and verify output shares, the Aggregators interact with one another
over ROUNDS
rounds. Prior to each round, each Aggregator constructs an
outbound message. Next, the sequence of outbound messages is combined into a
single message, called a "preparation message", or "prep message" for short.
(Each of the outbound messages are called "preparation-message shares", or
"prep shares" for short.) Finally, the preparation message is distributed to
the Aggregators to begin the next round.¶
An Aggregator begins the first round with its input share and it begins each subsequent round with the previous prep message. Its output in the last round is its output share and its output in each of the preceding rounds is a prep share.¶
This process involves a value called the "aggregation parameter" used to map the input shares to output shares. The Aggregators need to agree on this parameter before they can begin preparing the measurement shares for aggregation.¶
To facilitate the preparation process, a concrete VDAF implements the following methods:¶
vdaf.prep_init(verify_key: bytes, ctx: bytes, agg_id: int, agg_param:
AggParam, nonce: bytes, public_share: PublicShare, input_share: InputShare)
-> tuple[PrepState, PrepShare]
is the deterministic preparation-state
initialization algorithm run by each Aggregator to begin processing its input
share into an output share. Its inputs are the shared verification key
(verify_key
), the application context (ctx
), the Aggregator's unique
identifier (agg_id
), the aggregation parameter (agg_param
), the nonce
provided by the environment (nonce
, see Figure 7), the public share
(public_share
), and one of the input shares generated by the Client
(input_share
). Its output is the Aggregator's initial preparation state and
initial prep share.¶
It is up to the high level protocol in which the VDAF is used to arrange for the distribution of the verification key prior to generating and processing reports. (See Section 9 for details.)¶
Protocols MUST ensure that public share consumed by each of the Aggregators is identical. This is security critical for VDAFs such as Poplar1.¶
Pre-conditions:¶
vdaf.prep_next(ctx: bytes, prep_state: PrepState, prep_msg: PrepMessage) ->
tuple[PrepState, PrepShare] | OutShare
is the deterministic
preparation-state update algorithm run by each Aggregator. It updates the
Aggregator's preparation state (prep_state
) and returns either its next
preparation state and its message share for the current round or, if this is
the last round, its output share. An exception is raised if a valid output
share could not be recovered. The input of this algorithm is the inbound
preparation message.¶
vdaf.prep_shares_to_prep(ctx: bytes, agg_param: AggParam, prep_shares:
list[PrepShare]) -> PrepMessage
is the deterministic preparation-message
pre-processing algorithm. It combines the prep shares generated by the
Aggregators in the previous round into the prep message consumed by each in
the next round.¶
In effect, each Aggregator moves through a linear state machine with ROUNDS
states. The Aggregator enters the first state on using the initialization
algorithm, and the update algorithm advances the Aggregator to the next state.
Thus, in addition to defining the number of rounds (ROUNDS
), a VDAF instance
defines the state of the Aggregator after each round.¶
The preparation-state update accomplishes two tasks: recovery of output shares from the input shares and ensuring that the recovered output shares are valid. The abstraction boundary is drawn so that an Aggregator only recovers an output share if it is deemed valid (at least, based on the Aggregator's view of the protocol). Another way to draw this boundary would be to have the Aggregators recover output shares first, then verify that they are valid. However, this would allow the possibility of misusing the API by, say, aggregating an invalid output share. Moreover, in protocols like Prio+ [AGJOP21] based on oblivious transfer, it is necessary for the Aggregators to interact in order to recover aggregatable output shares at all.¶
Similar to DAFs (see Section 4.3), VDAFs MAY impose
restrictions for input shares and aggregation parameters. Protocols using a VDAF
MUST ensure that for each input share and aggregation parameter agg_param
, the
preparation phase (including vdaf.prep_init
, vdaf.prep_next
, and
vdaf.prep_shares_to_prep
; see Section 5.2) is only called if
vdaf.is_valid(agg_param, previous_agg_params)
returns True, where
previous_agg_params
contains all aggregation parameters that have previously
been used with the same input share.¶
VDAFs MUST implement the following function:¶
vdaf.is_valid(agg_param: AggParam, previous_agg_params: list[AggParam]) ->
bool
: checks if the agg_param
is compatible with all elements of
previous_agg_params
.¶
VDAF Aggregation is identical to DAF Aggregation (cf. Section 4.4):¶
vdaf.aggregate(agg_param: AggParam, out_shares: list[OutShare]) -> AggShare
is the deterministic aggregation algorithm. It is run by each Aggregator over
the output shares it has computed for a batch of measurements.¶
The data flow for this stage is illustrated in Figure 3. Here again, we have the aggregation algorithm in a "one-shot" form, where all shares for a batch are provided at the same time. VDAFs typically also support a "streaming" form, where shares are processed one at a time.¶
VDAF Unsharding is identical to DAF Unsharding (cf. Section 4.5):¶
vdaf.unshard(agg_param: AggParam, agg_shares: list[AggShare],
num_measurements: int) -> AggResult
is run by the Collector in order to
compute the aggregate result from the Aggregators' shares. The length of
agg_shares
MUST be SHARES
. num_measurements
is the number of
measurements that contributed to each of the aggregate shares. This algorithm
is deterministic.¶
Secure execution of a VDAF involves simulating the following procedure.¶
The inputs to this algorithm are the aggregation parameter, a list of measurements, and a nonce for each measurement. This document does not specify how the nonces are chosen, but security requires that the nonces be unique. See Section 9 for details. As explained in Section 4.6, the secure execution of a VDAF requires the application to instantiate secure channels between each of the protocol participants.¶
In each round of preparation, each Aggregator writes a prep share to some
broadcast channel, which is then processed into the prep message using the
public prep_shares_to_prep()
algorithm and broadcast to the Aggregators to
start the next round. In this section we describe some approaches to realizing
this broadcast channel functionality in protocols that use VDAFs.¶
The state machine of each Aggregator is shown in Figure 8.¶
State transitions are made when the state is acted upon by the host's local
inputs and/or messages sent by the peers. The initial state is Start
. The
terminal states are Rejected
, which indicates that the report cannot be
processed any further, and Finished(out_share)
, which indicates that the
Aggregator has recovered an output share out_share
.¶
class State: pass class Start(State): pass class Continued(State, Generic[PrepState]): def __init__(self, prep_state: PrepState, prep_round: int): self.prep_state = prep_state self.prep_round = prep_round def __eq__(self, other: object) -> bool: return isinstance(other, Continued) and \ self.prep_state == other.prep_state and \ self.prep_round == other.prep_round class Finished(State, Generic[OutShare]): def __init__(self, out_share: OutShare): self.out_share = out_share def __eq__(self, other: object) -> bool: return isinstance(other, Finished) and \ self.out_share == other.out_share class Rejected(State): pass¶
The methods described in this section are defined in terms of opaque byte
strings. A compatible Vdaf
MUST specify methods for encoding public shares,
input shares, prep shares, prep messages, and aggregation parameters.¶
Implementations of Prio3 and Poplar1 MUST use the encoding scheme specified in Section 7.2.7 and Section 8.2.6 respectively.¶
For VDAFs with precisely two Aggregators (i.e., SHARES == 2
), the following
"ping pong" communication pattern can be used. It is compatible with any
request/response transport protocol, such as HTTP.¶
Let us call the initiating party the "Leader" and the responding party the "Helper". The high-level idea is that the Leader and Helper will take turns running the computation locally until input from their peer is required:¶
For a 1-round VDAF (e.g., Prio3 (Section 7)), the Leader sends its prep share to the Helper, who computes the prep message locally, computes its output share, then sends the prep message to the Leader. Preparation requires just one round trip between the Leader and the Helper.¶
For a 2-round VDAF (e.g., Poplar1 (Section 8)), the Leader sends its first-round prep share to the Helper, who replies with the first-round prep message and its second-round prep share. In the next request, the Leader computes its second-round prep share locally, computes its output share, and sends the second-round prep message to the Helper. Finally, the Helper computes its own output share.¶
In general, each request includes the Leader's prep share for the previous round and/or the prep message for the current round; correspondingly, each response consists of the prep message for the current round and the Helper's prep share for the next round.¶
The Aggregators proceed in this ping-ponging fashion until a step of the
computation fails (indicating the report is invalid and should be rejected) or
preparation is completed. All told there there are ceil((ROUNDS+1)/2)
requests sent.¶
Each message in the ping-pong protocol is structured as follows (expressed in TLS syntax as defined in Section 3 of [RFC8446]):¶
enum { initialize(0), continue(1), finish(2), (255) } MessageType; struct { MessageType type; select (Message.type) { case initialize: opaque prep_share<0..2^32-1>; case continue: opaque prep_msg<0..2^32-1>; opaque prep_share<0..2^32-1>; case finish: opaque prep_msg<0..2^32-1>; }; } Message;¶
These messages are used to transition between the states described in
Section 5.7. The Leader's initial transition is computed with the
following method, implemented on Vdaf
:¶
def ping_pong_leader_init( self, vdaf_verify_key: bytes, ctx: bytes, agg_param: bytes, nonce: bytes, public_share: bytes, input_share: bytes) -> tuple[State, Optional[bytes]]: """Called by the leader to initialize ping-ponging.""" try: (prep_state, prep_share) = self.prep_init( vdaf_verify_key, ctx, 0, self.decode_agg_param(agg_param), nonce, self.decode_public_share(public_share), self.decode_input_share(0, input_share), ) encoded_prep_share = self.encode_prep_share(prep_share) return ( Continued(prep_state, 0), encode(0, encoded_prep_share), # initialize ) except: return (Rejected(), None)¶
The output is the State
to which the Leader has transitioned and an encoded
Message
. If the Leader's state is Rejected
, then processing halts.
Otherwise, if the state is Continued
, then processing continues. Function
encode
is used to encode the outbound message, here the initialize
variant
(hence 0
).¶
The Leader sends the outbound message to the Helper. The Helper's initial transition is computed using the following procedure:¶
def ping_pong_helper_init( self, vdaf_verify_key: bytes, ctx: bytes, agg_param: bytes, nonce: bytes, public_share: bytes, input_share: bytes, inbound: bytes) -> tuple[State, Optional[bytes]]: """ Called by the helper in response to the leader's initial message. """ try: (prep_state, prep_share) = self.prep_init( vdaf_verify_key, ctx, 1, self.decode_agg_param(agg_param), nonce, self.decode_public_share(public_share), self.decode_input_share(1, input_share), ) (inbound_type, inbound_items) = decode(inbound) if inbound_type != 0: # initialize return (Rejected(), None) encoded_prep_share = inbound_items[0] prep_shares = [ self.decode_prep_share(prep_state, encoded_prep_share), prep_share, ] return self.ping_pong_transition( ctx, self.decode_agg_param(agg_param), prep_shares, prep_state, 0, ) except: return (Rejected(), None)¶
Procedure decode
decodes the inbound message and returns the MessageType
variant (initialize
, continue
, or finalize
) and the sequence of fields.
Procedure ping_pong_transition
takes in the prep shares, combines them into
the prep message, and computes the next prep state of the caller:¶
def ping_pong_transition( self, ctx: bytes, agg_param: AggParam, prep_shares: list[PrepShare], prep_state: PrepState, prep_round: int) -> tuple[State, bytes]: prep_msg = self.prep_shares_to_prep(ctx, agg_param, prep_shares) encoded_prep_msg = self.encode_prep_msg(prep_msg) out = self.prep_next(ctx, prep_state, prep_msg) if prep_round+1 == self.ROUNDS: return ( Finished(out), encode(2, encoded_prep_msg), # finalize ) (prep_state, prep_share) = cast( tuple[PrepState, PrepShare], out) encoded_prep_share = self.encode_prep_share(prep_share) return ( Continued(prep_state, prep_round+1), encode(1, encoded_prep_msg, encoded_prep_share) # continue )¶
The output is the State
to which the Helper has transitioned and an encoded
Message
. If the Helper's state is Finished
or Rejected
, then processing
halts. Otherwise, if the state is Continued
, then processing continues.¶
Next, the Helper sends the outbound message to the Leader. The Leader computes
its next state transition using the function ping_pong_leader_continued
:¶
def ping_pong_leader_continued( self, ctx: bytes, agg_param: bytes, state: State, inbound: bytes, ) -> tuple[State, Optional[bytes]]: """ Called by the leader to start the next step of ping-ponging. """ return self.ping_pong_continued( True, ctx, agg_param, state, inbound) def ping_pong_continued( self, is_leader: bool, ctx: bytes, agg_param: bytes, state: State, inbound: bytes, ) -> tuple[State, Optional[bytes]]: try: if not isinstance(state, Continued): return (Rejected(), None) prep_round = state.prep_round (inbound_type, inbound_items) = decode(inbound) if inbound_type == 0: # initialize return (Rejected(), None) encoded_prep_msg = inbound_items[0] prep_msg = self.decode_prep_msg( state.prep_state, encoded_prep_msg, ) out = self.prep_next(ctx, state.prep_state, prep_msg) if prep_round+1 < self.ROUNDS and \ inbound_type == 1: # continue (prep_state, prep_share) = cast( tuple[PrepState, PrepShare], out) encoded_prep_share = inbound_items[1] prep_shares = [ self.decode_prep_share( prep_state, encoded_prep_share, ), prep_share, ] if is_leader: prep_shares.reverse() return self.ping_pong_transition( ctx, self.decode_agg_param(agg_param), prep_shares, prep_state, prep_round+1, ) elif prep_round+1 == self.ROUNDS and \ inbound_type == 2: # finish return (Finished(out), None) else: return (Rejected(), None) except: return (Rejected(), None)¶
If the Leader's state is Finished
or Rejected
, then processing halts.
Otherwise, the Leader sends the outbound message to the Helper. The Helper
computes its next state transition using the function
ping_pong_helper_continued
:¶
def ping_pong_helper_continued( self, ctx: bytes, agg_param: bytes, state: State, inbound: bytes, ) -> tuple[State, Optional[bytes]]: """Called by the helper to continue ping-ponging.""" return self.ping_pong_continued( False, ctx, agg_param, state, inbound)¶
They continue in this way until processing halts. Note that, depending on the
number of rounds of preparation that are required, there may be one more
message to send before the peer can also finish processing (i.e., outbound !=
None
).¶
The ping-pong topology of the previous section is only suitable for VDAFs involving exactly two Aggregators. If the VDAF supports more than two Aggregators, then the star topology described in this section can be used instead.¶
We again designate an Aggregator to initiate the computation. We refer to this Aggregator as the Leader and to all other Aggregators as Helpers.¶
At the start of each round, the Leader requests from each Helper its prep
share. After gathering each of the prep shares, the Leader computes the next
prep message (via prep_shares_to_prep()
) and broadcasts it to the Helpers. At
this point, each Aggregator runs prep_next()
locally to either recover an
output share or, if more rounds of preparation are required, compute its updated state
and prep share. If more are required, then the Helper responds to the broadcast
message with its next prep share.¶
The Aggregators proceed in this way until each recovers an output share or some step of the computation fails.¶
This section describes the primitives that are common to the VDAFs specified in this document.¶
Both Prio3 and Poplar1 use finite fields of prime order. Finite field
elements are represented by a class Field
with the following associated
parameters:¶
MODULUS: int
is the prime modulus that defines the field.¶
ENCODED_SIZE: int
is the number of bytes used to encode a field element
as a byte string.¶
A concrete Field
also implements the following class methods:¶
Field.zeros(length: int) -> list[Self]
returns a vector of
zeros.¶
Pre-conditions:¶
length
MUST be greater than or equal 0
.¶
Post-conditions:¶
The length of the output MUST be length
.¶
Field.rand_vec(length: int) -> list[Self]
returns a vector of
random field elements. Same pre- and post-conditions as for Field.zeros()
.¶
A field element is an instance of a concrete Field
. The concrete class defines
the usual arithmetic operations on field elements. In addition, it defines the
following instance method for converting a field element to an unsigned integer:¶
elem.as_unsigned() -> int
returns the integer representation of
field element elem
.¶
Likewise, each concrete Field
implements a constructor for converting an
unsigned integer into a field element:¶
Field(integer: int)
returns integer
represented as a field element.
The value of integer
MUST be non-negative and less than Field.MODULUS
.¶
Each concrete Field
has two derived class methods, one for encoding
a vector of field elements as a byte string and another for decoding a vector of
field elements.¶
Finally, Field
implements the following methods for representing a value as
a sequence of field elements, each of which represents a bit of the input.¶
The following auxiliary functions on vectors of field elements are used in the remainder of this document. Note that an exception is raised by each function if the operands are not the same length.¶
Some VDAFs require fields that are suitable for efficient computation of the number theoretic transform (NTT) [SML24], as this allows for fast polynomial interpolation. (One example is Prio3 (Section 7) when instantiated with the FLP of Section 7.3.3.) Specifically, a field is said to be "NTT-friendly" if, in addition to satisfying the interface described in Section 6.1, it implements the following method:¶
Field.gen() -> Field
returns the generator of a large subgroup of the
multiplicative group. To be NTT-friendly, the order of this subgroup MUST be a
power of 2. In addition, the size of the subgroup dictates how large
interpolated polynomials can be. It is RECOMMENDED that a generator is chosen
with order at least 2^20
.¶
NTT-friendly fields also define the following parameter:¶
GEN_ORDER: int
is the order of a multiplicative subgroup generated by
Field.gen()
.¶
The tables below define finite fields used in the remainder of this document.¶
Parameter | Field64 | Field128 | Field255 |
---|---|---|---|
MODULUS | 2^32 * 4294967295 + 1 | 2^66 * 4611686018427387897 + 1 | 2^255 - 19 |
ENCODED_SIZE | 8 | 16 | 32 |
Generator | 7^4294967295 | 7^4611686018427387897 | n/a |
GEN_ORDER | 2^32 | 2^66 | n/a |
VDAFs in this specification use extendable output functions (XOFs) to extract short, fixed-length strings we call "seeds" from long input strings and expand seeds into long output strings. We specify a single interface that is suitable for both purposes.¶
XOFs are defined by a class Xof
with the following associated parameter and
methods:¶
SEED_SIZE: int
is the size (in bytes) of a seed.¶
Xof(seed: bytes, dst: bytes, binder: bytes)
constructs an instance of Xof
from the given seed, domain separation tag, and binder string. (See below for
definitions of these.) The length of the seed will typically be SEED_SIZE
,
but some XOFs may support multiple seed sizes. The seed MUST be generated
securely (i.e., it is either the output of a CSPRNG or a previous invocation
of the XOF).¶
xof.next(length: int)
returns the next length
bytes of output of
xof
.¶
Each Xof
has three derived methods. The first is used to derive a fresh seed
from an existing one. The second is used to compute a sequence of field
elements. The third is a convenience method to construct an Xof
from a seed,
domain separation tag, and binder string, and then use it to compute a sequence
of field elements.¶
This section describes XofTurboShake128, an XOF based on the TurboSHAKE128
[TurboSHAKE]. (RFC EDITOR: Update this
reference to the RFC for draft-irtf-cfrg-kangarootwelve once published.) This
XOF is RECOMMENDED for all use cases within VDAFs. The length of the domain
separation string dst
passed to XofTurboShake128 MUST NOT exceed 65535 bytes.¶
While XofTurboShake128 as described above can be securely used in all cases
where a XOF is needed in the VDAFs described in this document, there are some
cases where a more efficient instantiation based on fixed-key AES is possible.
For now, this is limited to the XOF used inside the Idpf Section 8.1
implementation in Poplar1 Section 8.3. It is NOT RECOMMENDED to use this
XOF anywhere else. The length of the domain separation string dst
passed to
XofFixedKeyAes128 MUST NOT exceed 65535 bytes. See Section 9 for a more
detailed discussion.¶
class XofFixedKeyAes128(Xof): """ XOF based on a circular collision-resistant hash function from fixed-key AES. """ # Associated parameters SEED_SIZE = 16 def __init__(self, seed: bytes, dst: bytes, binder: bytes): if len(seed) != self.SEED_SIZE: raise ValueError("incorrect seed size") self.length_consumed = 0 # Use TurboSHAKE128 to derive a key from the binder string # and domain separation tag. Note that the AES key does not # need to be kept secret from any party. However, when used # with an IDPF, we require the binder to be a random nonce. # # Implementation note: this step can be cached across XOF # evaluations with many different seeds. dst_length = to_le_bytes(len(dst), 2) self.fixed_key = TurboSHAKE128( dst_length + dst + binder, 2, 16, ) self.seed = seed def next(self, length: int) -> bytes: offset = self.length_consumed % 16 new_length = self.length_consumed + length block_range = range( self.length_consumed // 16, new_length // 16 + 1 ) self.length_consumed = new_length hashed_blocks = [ self.hash_block(xor(self.seed, to_le_bytes(i, 16))) for i in block_range ] return concat(hashed_blocks)[offset:offset+length] def hash_block(self, block: bytes) -> bytes: """ The multi-instance tweakable circular correlation-robust hash function of [GKWWY20] (Section 4.2). The tweak here is the key that stays constant for all XOF evaluations of the same Client, but differs between Clients. Function `AES128(key, block)` is the AES-128 blockcipher. """ lo, hi = block[:8], block[8:] sigma_block = concat([hi, xor(hi, lo)]) return xor(AES128(self.fixed_key, sigma_block), sigma_block)¶
XOFs are used to map a seed to a finite domain, e.g., a fresh seed or a vector of field elements. To ensure domain separation, the derivation is needs to be bound to some distinguished domain separation tag. The domain separation tag encodes the following values:¶
The document version (i.e.,VERSION
);¶
The "class" of the algorithm using the output (e.g., VDAF);¶
A unique identifier for the algorithm; and¶
Some indication of how the output is used (e.g., for deriving the measurement shares in Prio3 Section 7).¶
The following algorithm is used in the remainder of this document in order to format the domain separation tag:¶
def format_dst(algo_class: int, algo: int, usage: int) -> bytes: """ Format XOF domain separation tag for use within a (V)DAF. Pre-conditions: - `algo_class` in `range(0, 2 ** 8)` - `algo` in `range(0, 2 ** 32)` - `usage` in `range(0, 2 ** 16)` """ return concat([ to_be_bytes(VERSION, 1), to_be_bytes(algo_class, 1), to_be_bytes(algo, 4), to_be_bytes(usage, 2), ])¶
It is also sometimes necessary to bind the output to some ephemeral value that multiple parties need to agree on. We call this input the "binder string".¶
This section describes Prio3, a VDAF for Prio [CGB17]. Prio is suitable for a wide variety of aggregation functions, including (but not limited to) sum, mean, standard deviation, estimation of quantiles (e.g., median), and linear regression. In fact, the scheme described in this section is compatible with any aggregation function that has the following structure:¶
Each measurement is encoded as a vector over some finite field.¶
Measurement validity is determined by an arithmetic circuit evaluated over the encoded measurement. (An "arithmetic circuit" is a function comprised of arithmetic operations in the field.) The circuit's output is a single field element: if zero, then the measurement is said to be "valid"; otherwise, if the output is non-zero, then the measurement is said to be "invalid".¶
The aggregate result is obtained by summing up the encoded measurement vectors and computing some function of the sum.¶
At a high level, Prio3 distributes this computation as follows. Each Client first shards its measurement by first encoding it, then splitting the vector into secret shares and sending a share to each Aggregator. Next, in the preparation phase, the Aggregators carry out a multi-party computation to determine if their shares correspond to a valid measurement (as determined by the arithmetic circuit). This computation involves a "proof" of validity generated by the Client. Next, each Aggregator sums up its shares locally. Finally, the Collector sums up the aggregate shares and computes the aggregate result.¶
This VDAF does not have an aggregation parameter. Instead, the output share is derived from the measurement share by applying a fixed map. See Section 8 for an example of a VDAF that makes meaningful use of the aggregation parameter.¶
The core component of Prio3 is a "Fully Linear Proof (FLP)" system. Introduced by [BBCGGI19], the FLP encapsulates the functionality required for encoding and validating measurements. Prio3 can be thought of as a transformation of a particular class of FLPs into a VDAF.¶
The remainder of this section is structured as follows. The syntax for FLPs is described in Section 7.1. The generic transformation of an FLP into Prio3 is specified in Section 7.2. Next, a concrete FLP suitable for any validity circuit is specified in Section 7.3. Finally, instantiations of Prio3 for various types of measurements are specified in Section 7.4. Test vectors can be found in Appendix "Test Vectors".¶
Conceptually, an FLP is a two-party protocol executed by a prover and a verifier. In actual use, however, the prover's computation is carried out by the Client, and the verifier's computation is distributed among the Aggregators. The Client generates a "proof" of its measurement's validity and distributes shares of the proof to the Aggregators. Each Aggregator then performs some computation on its measurement share and proof share locally and sends the result to the other Aggregators. Combining the exchanged messages allows each Aggregator to decide if it holds a share of a valid measurement. (See Section 7.2 for details.)¶
As usual, we will describe the interface implemented by a concrete FLP in terms
of an abstract base class Flp
that specifies the set of methods and parameters
a concrete FLP must provide.¶
The parameters provided by a concrete FLP are listed in Table 4.¶
Parameter | Description |
---|---|
PROVE_RAND_LEN
|
Length of the prover randomness, the number of random field elements consumed by the prover when generating a proof |
QUERY_RAND_LEN
|
Length of the query randomness, the number of random field elements consumed by the verifier |
JOINT_RAND_LEN
|
Length of the joint randomness, the number of random field elements consumed by both the prover and verifier |
MEAS_LEN
|
Length of the encoded measurement (Section 7.1.1) |
OUTPUT_LEN
|
Length of the aggregatable output (Section 7.1.1) |
PROOF_LEN
|
Length of the proof |
VERIFIER_LEN
|
Length of the verifier message generated by querying the measurement and proof |
Measurement
|
Type of the measurement |
AggResult
|
Type of the aggregate result |
field
|
Class object for the field (Section 6.1) |
An FLP specifies the following algorithms for generating and verifying proofs of validity (encoding is described below in Section 7.1.1):¶
flp.prove(meas: list[F], prove_rand: list[F], joint_rand: list[F]) ->
list[F]
is the deterministic proof-generation algorithm run by the prover.
Its inputs are the encoded measurement, the "prover randomness" prove_rand
,
and the "joint randomness" joint_rand
. The prover randomness is used only
by the prover, but the joint randomness is shared by both the prover and
verifier.¶
flp.query(meas: list[F], proof: list[F], query_rand: list[F], joint_rand:
list[F], num_shares: int) -> list[F]
is the query-generation algorithm run
by the verifier. This is used to "query" the measurement and proof. The
result of the query (i.e., the output of this function) is called the
"verifier message". In addition to the measurement and proof, this algorithm
takes as input the query randomness query_rand
and the joint randomness
joint_rand
. The former is used only by the verifier. num_shares
specifies
how many shares were generated.¶
flp.decide(verifier: list[F]) -> bool
is the deterministic decision
algorithm run by the verifier. It takes as input the verifier message and
outputs a boolean indicating if the measurement from which it was generated
is valid.¶
Our application requires that the FLP is "fully linear" in the sense defined in [BBCGGI19]. As a practical matter, what this property implies is that, when run on a share of the measurement and proof, the query-generation algorithm outputs a share of the verifier message. Furthermore, the privacy property of the FLP system ensures that the verifier message reveals nothing about the measurement other than whether it is valid. Therefore, to decide if a measurement is valid, the Aggregators will run the query-generation algorithm locally, exchange verifier shares, combine them to recover the verifier message, and run the decision algorithm.¶
The query-generation algorithm includes a parameter num_shares
that specifies
the number of shares that were generated. If these data are not secret shared,
then num_shares == 1
. This parameter is useful for certain FLP constructions.
For example, the FLP in Section 7.3 is defined in terms of an arithmetic
circuit; when the circuit contains constants, it is sometimes necessary to
normalize those constants to ensure that the circuit's output, when run on a
valid measurement, is the same regardless of the number of shares.¶
An FLP is executed by the prover and verifier as follows:¶
The proof system is constructed so that, if meas
is valid, then run_flp(flp,
meas, 1)
always returns True
. On the other hand, if meas
is invalid, then
as long as joint_rand
and query_rand
are generated uniform randomly, the
output is False
with high probability. False positives are possible: there is
a small probability that a verifier accepts an invalid input as valid. An FLP
is said to be "sound" if this probability is sufficiently small. The soundness
of the FLP depends on a variety of parameters, like the length of the
input and the size of the field. See Section 7.3 for details.¶
Note that soundness of an FLP system is not the same as robustness for a VDAF In particular, soundness of the FLP is necessary, but insufficient for robusntess of Prio3 (Section 7). See Section 9.6 for details.¶
We remark that [BBCGGI19] defines a much larger class of fully linear proof systems than we consider here. In particular, what is called an "FLP" here is called a 1.5-round, public-coin, interactive oracle proof system in their paper.¶
The type of measurement being aggregated is defined by the FLP. Hence, the FLP also specifies a method of encoding raw measurements as a vector of field elements:¶
flp.encode(measurement: Measurement) -> list[F]
encodes a raw measurement
as a vector of field elements. The return value MUST be of length MEAS_LEN
.¶
For some FLPs, the encoded measurement also includes redundant field elements
that are useful for checking the proof, but which are not needed after the
proof has been checked. An example is the "integer sum" data type from
[CGB17] in which an integer in range(2**k)
is encoded as a vector of k
field elements, each representing a bit of the integer (this type is also
defined in Section 7.4.2). After consuming this vector, all that is needed is
the integer it represents. Thus the FLP defines an algorithm for truncating the
encoded measurement to the length of the aggregated output:¶
flp.truncate(meas: list[F]) -> list[F]
maps an encoded measurement (e.g.,
the bit-encoding of the measurement) to an aggregatable output (e.g., the
singleton vector containing the measurement). The length of the input MUST be
MEAS_LEN
and the length of the output MUST be OUTPUT_LEN
.¶
Once the aggregate shares have been computed and combined together, their sum can be converted into the aggregate result. This could be a projection from the FLP's field to the integers, or it could include additional post-processing.¶
flp.decode(output: list[F], num_measurements: int) -> AggResult
maps a sum
of aggregate shares to an aggregate result.¶
Pre-conditions:¶
We remark that, taken together, these three functionalities correspond roughly to the notion of "Affine-aggregatable encodings (AFEs)" from [CGB17].¶
It is sometimes desirable to generate and verify multiple independent proofs for the same input. First, this improves the soundness of the proof system without having to change any of its parameters. Second, it allows a smaller field to be used (e.g., replace Field128 with Field64, see Section 7.3) without sacrificing soundness. Generally, choosing a smaller field can significantly reduce communication cost. (This is a trade-off, of course, since generating and verifying more proofs requires more time.) Given these benefits, this feature is implemented by Prio3 (Section 7).¶
To generate these proofs for a specific measurement, the prover calls
flp.prove
multiple times, each time using an independently generated prover
and joint randomness string. The verifier checks each proof independently, each
time with an independently generated query randomness string. It accepts the
measurement only if all the decision algorithm accepts on each proof.¶
See Section 9.6 below for discussions on choosing the right number of proofs.¶
This section specifies Prio3
, an implementation of the Vdaf
interface
(Section 5). It has three generic parameters: an NttField ({{ntt-field}}), an
Flp ({{flp}}) and a
Xof ({{xof}}). It also has an associated constant,
PROOFS, with a value in
range(1, 256)`, denoting the number of FLPs
generated by the Client (Section 7.1.2).¶
The associated constants and types required by the Vdaf
interface are
defined in Table 5. The methods required for sharding,
preparation, aggregation, and unsharding are described in the remaining
subsections. These methods refer to constants enumerated in
Table 6.¶
Parameter | Value |
---|---|
VERIFY_KEY_SIZE
|
Xof.SEED_SIZE
|
RAND_SIZE
|
Xof.SEED_SIZE * SHARES if flp.JOINT_RAND_LEN == 0 else 2 * Xof.SEED_SIZE * SHARES
|
NONCE_SIZE
|
16
|
ROUNDS
|
1
|
SHARES
|
in [2, 256)
|
Measurement
|
Flp.Measurement
|
AggParam
|
None
|
PublicShare
|
Optional[list[bytes]]
|
InputShare
|
tuple[list[F], list[F], Optional[bytes]] | tuple[bytes, Optional[bytes]]
|
OutShare
|
list[F]
|
AggShare
|
list[F]
|
AggResult
|
Flp.AggResult
|
PrepState
|
tuple[list[F], Optional[bytes]]
|
PrepShare
|
tuple[list[F], Optional[bytes]]
|
PrepMessage
|
Optional[bytes]
|
Variable | Value |
---|---|
USAGE_MEAS_SHARE: int
|
1 |
USAGE_PROOF_SHARE: int
|
2 |
USAGE_JOINT_RANDOMNESS: int
|
3 |
USAGE_PROVE_RANDOMNESS: int
|
4 |
USAGE_QUERY_RANDOMNESS: int
|
5 |
USAGE_JOINT_RAND_SEED: int
|
6 |
USAGE_JOINT_RAND_PART: int
|
7 |
This section describes the process of recovering output shares from the input shares. The high-level idea is that each Aggregator first queries its measurement and share of proof(s) locally, then exchanges its share of verifier(s) with the other Aggregators. The shares of verifier(s) are then combined into the verifier message(s) used to decide whether to accept.¶
In addition, for FLPs that require joint randomness, the Aggregators must ensure that they have all used the same joint randomness for the query-generation algorithm. To do so, they collectively re-derive the joint randomness from their measurement shares just as the Client did during sharding.¶
In order to avoid extra round of communication, the Client sends each Aggregator a "hint" consisting of the joint randomness parts. This leaves open the possibility that the Client cheated by, say, forcing the Aggregators to use joint randomness that biases the proof check procedure some way in its favor. To mitigate this, the Aggregators also check that they have all computed the same joint randomness seed before accepting their output shares. To do so, they exchange their parts of the joint randomness along with their shares of verifier(s).¶
Implementation note: the preparation state for Prio3 includes the output share that will be released once preparation is complete. In some situations, it may be necessary for the Aggregator to encode this state as bytes and store it for retrieval later on. For all but the first Aggregator, it is possible to save storage by storing the measurement share rather than output share itself. It is relatively inexpensive to expand this seed into the input share, then truncate the input share to get the output share.¶
The definitions of constants and a few auxiliary functions are defined in Section 7.2.6.¶
Every input share MUST only be used once, regardless of the aggregation parameters used.¶
Aggregating a set of output shares is simply a matter of adding up the vectors element-wise.¶
To unshard a set of aggregate shares, the Collector first adds up the vectors element-wise. It then converts each element of the vector into an integer.¶
This section defines a number of auxiliary functions referenced by the main algorithms for Prio3 in the preceding sections.¶
The following methods are called by the sharding and preparation algorithms.¶
def helper_meas_share( self, ctx: bytes, agg_id: int, k_share: bytes) -> list[F]: return self.xof.expand_into_vec( self.flp.field, k_share, self.domain_separation_tag(USAGE_MEAS_SHARE, ctx), byte(agg_id), self.flp.MEAS_LEN, ) def helper_proofs_share( self, ctx: bytes, agg_id: int, k_share: bytes) -> list[F]: return self.xof.expand_into_vec( self.flp.field, k_share, self.domain_separation_tag(USAGE_PROOF_SHARE, ctx), byte(self.PROOFS) + byte(agg_id), self.flp.PROOF_LEN * self.PROOFS, ) def expand_input_share( self, ctx: bytes, agg_id: int, input_share: Prio3InputShare[F]) -> tuple[ list[F], list[F], Optional[bytes]]: if agg_id > 0: assert len(input_share) == 2 (k_share, k_blind) = input_share meas_share = self.helper_meas_share(ctx, agg_id, k_share) proofs_share = self.helper_proofs_share(ctx, agg_id, k_share) else: assert len(input_share) == 3 (meas_share, proofs_share, k_blind) = input_share return (meas_share, proofs_share, k_blind) def prove_rands(self, ctx: bytes, k_prove: bytes) -> list[F]: return self.xof.expand_into_vec( self.flp.field, k_prove, self.domain_separation_tag(USAGE_PROVE_RANDOMNESS, ctx), byte(self.PROOFS), self.flp.PROVE_RAND_LEN * self.PROOFS, ) def query_rands( self, verify_key: bytes, ctx: bytes, nonce: bytes) -> list[F]: return self.xof.expand_into_vec( self.flp.field, verify_key, self.domain_separation_tag(USAGE_QUERY_RANDOMNESS, ctx), byte(self.PROOFS) + nonce, self.flp.QUERY_RAND_LEN * self.PROOFS, ) def joint_rand_part( self, ctx: bytes, agg_id: int, k_blind: bytes, meas_share: list[F], nonce: bytes) -> bytes: return self.xof.derive_seed( k_blind, self.domain_separation_tag(USAGE_JOINT_RAND_PART, ctx), byte(agg_id) + nonce + self.flp.field.encode_vec(meas_share), ) def joint_rand_seed(self, ctx: bytes, k_joint_rand_parts: list[bytes]) -> bytes: """Derive the joint randomness seed from its parts.""" return self.xof.derive_seed( zeros(self.xof.SEED_SIZE), self.domain_separation_tag(USAGE_JOINT_RAND_SEED, ctx), concat(k_joint_rand_parts), ) def joint_rands(self, ctx: bytes, k_joint_rand_seed: bytes) -> list[F]: """Derive the joint randomness from its seed.""" return self.xof.expand_into_vec( self.flp.field, k_joint_rand_seed, self.domain_separation_tag(USAGE_JOINT_RANDOMNESS, ctx), byte(self.PROOFS), self.flp.JOINT_RAND_LEN * self.PROOFS, )¶
This section defines serialization formats for messages exchanged over the network while executing Prio3. It is RECOMMENDED that implementations provide serialization methods for them.¶
Message structures are defined following Section 3 of [RFC8446]). In the
remainder we use S
as an alias for prio3.xof.SEED_SIZE
and F
as an alias
for prio3.field.ENCODED_SIZE
. XOF seeds are represented as follows:¶
opaque Prio3Seed[S];¶
Field elements are encoded in little-endian byte order (as defined in Section 6.1) and represented as follows:¶
opaque Prio3Field[F];¶
When joint randomness is not used, the prep message is the empty string. Otherwise the prep message consists of the joint randomness seed computed by the Aggregators:¶
struct { Prio3Seed k_joint_rand; } Prio3PrepMessageWithJointRand;¶
Aggregate shares are structured as follows:¶
struct { Prio3Field agg_share[F * prio3.flp.OUTPUT_LEN]; } Prio3AggShare;¶
This section describes an FLP based on the construction from [BBCGGI19], Section 4.2. We begin in Section 7.3.1 with an overview of their proof system and some extensions to it. The construction is specified in Section 7.3.3.¶
Conventional zero-knowledge proof systems involve two parties:¶
The prover, who holds a measurement and generates a proof of the measurement's validity¶
The verifier who holds an encryption of, or commitment to, the measurement and checks the proof¶
Our proof system is much the same, except the verifier is split across multiple Aggregators, each of which has a secret share of the measurement rather than an encryption of it.¶
Validity is defined in terms of an arithmetic circuit evaluated over the measurement. The inputs to this circuit are elements of a finite field that comprise the encoded measurement; the gates of the circuit are multiplication, addition, and subtraction operations; and the output of the circuit is a single field element. If the value is zero, then the measurement is deemed valid; otherwise, if the output is non-zero, then the measurement is deemed invalid.¶
For example, the simplest circuit specified in this document is the following (Section 7.4.1):¶
C(x) = x * (x-1)¶
This circuit contains one subtraction gate (x - 1
) and one multiplication
gate (x * (x - 1)
). Observe that C(x) = 0
if and only if x in range(2)
.¶
Our goal is to allow each Aggregator, who holds a secret share of x
, to
correctly compute a secret share of C(x)
. This allows the Aggregators to
determine validity by combining their shares of the output.¶
Suppose for a moment that the validity circuit C
is an affine arithmetic
circuit, meaning its only operations are addition, subtraction, and
multiplication-by-constant. (The circuit above is non-affine because it
contains a multiplication gate with two non-constant inputs.) Then each
Aggregator can compute its share locally, since¶
C(x_shares[0] + ... + x_shares[SHARES-1]) = C(x_shares[0]) + ... + C(x_shares[SHARES-1])¶
(Note that, for this equality to hold, it is necessary to scale any addition of
a constant in the circuit by 1/SHARES
.) However, this is not the case if C
contains multiplication gates with two non-constant inputs. Thus our goal is to
transform these multiplication gates into computations on secret shared data
that each Aggregator can perform locally.¶
The key idea is to have the prover construct a polynomial p
such that p(j)
is equal to the output of the j
-th multiplication gate. Polynomial evaluation
is fully linear, which means the coefficients of the polynomial can be secret
shared in a way that allows each Aggregator to compute a share of p(j)
for
any j
. These intermediate results can then be combined with the affine
arithmetic operations of the validity circuit to produce the final output.¶
Applying this idea to the example circuit C
above:¶
The Client, given its measurement x
, constructs the lowest degree
polynomial p
for which p(0) = s
and p(1) = x * (x - 1)
, where s
is a
random blinding value generated by the Client. (The blinding value is to
protect the privacy of the measurement.) It sends secret shares of x
and
the coefficients of p
to each of the Aggregators.¶
Each Aggregator locally computes and broadcasts its share of p(1)
, which
is equal to its share of C(x)
.¶
In fact, our FLP is slightly more general than this. We can replace the multiplication gate with any non-affine sub-circuit and apply the same idea. For example, in Section 7.4.2, the validity circuit uses the following sub-circuit multiple times:¶
Range2(x) = x * (x - 1) = x^2 - x¶
(This is the same functionality computed by the example circuit C
above.)
Here again we can interpolate the lowest degree polynomial p
for which p(j)
is the value of the j
-th call to Range2
in the validity circuit. Each
validity circuit defines a sub-circuit that encapsulates its non-affine
arithmetic operations. We refer to this sub-circuit as the "gadget".¶
This idea provides the needed functionality, but it has one more problem: it is
possible for a malicious Client to produce a gadget polynomial p
that would
result in C(x)
being computed incorrectly, potentially resulting in an
invalid measurement being accepted. To prevent this, the Aggregators perform a
probabilistic test to check that the gadget polynomial was constructed
properly. This "gadget test", and the procedure for constructing the
polynomial, are described in detail in Section 7.3.3.¶
The FLP described in {#flp-bbcggi19-construction} extends the proof system of [BBCGGI19], Section 4.2 in a few ways.¶
First, the validity circuit in our construction includes an additional, random
input (this is the "joint randomness" derived from the measurement shares in
Prio3; see Section 7.2). This allows for circuit optimizations that
trade a small soundness error for a shorter proof. For example, consider a
circuit that recognizes the set of length-N
vectors for which each element is
either one or zero. A deterministic circuit could be constructed for this
language, but it would involve a large number of multiplications that would
result in a large proof. (See the discussion in [BBCGGI19], Section 5.2 for
details). A much shorter proof can be constructed for the following randomized
circuit:¶
C(x, r) = r * Range2(x[0]) + ... + r^N * Range2(x[N-1])¶
(Note that this is a special case of [BBCGGI19], Theorem 5.2.) Here x
is
the length-N
input and r
is a random field element. The gadget circuit
Range2
is the "range-check" polynomial described above, i.e., Range2(x) =
x^2 - x
. The idea is that, if x
is valid (i.e., each x[j]
is in
range(2)
), then the circuit will evaluate to 0 regardless of the value of
r
; but if some x[j]
is not in range(2)
, then output will be non-zero with
high probability.¶
The second extension implemented by our FLP allows the validity circuit to contain multiple gadget types. (This generalization was suggested in [BBCGGI19], Remark 4.5.) This provides additional flexibility for designing circuits by allowing multiple, non-affine sub-circuits. For example, the following circuit is allowed:¶
C(meas, r) = r * Range2(meas[0]) + ... + r^L * Range2(meas[L-1]) + \ r^(L+1) * Range3(meas[L]) + ... + r^N * Range3(meas[N-1])¶
where Range3(x) = x^3 - 3x^2 + 2x
. This circuit checks that the first L
inputs are in range(2)
and the last N-L
inputs are in range(3)
. The
same circuit can be expressed using a simpler gadget, namely multiplication,
but the resulting proof would be longer.¶
Third, rather than interpolate the gadget polynomial at inputs 1
, 2
, ...,
j
, ..., where j
is the j
-th invocation of the gadget, we use powers of
alpha
, where alpha
is a root of unity for the field. This allows us to
construct each gadget polynomial via the number theoretic transform [SML24],
which is far more efficient than generic formulas. This requires the field to
be "NTT-friendly" as defined in Section 6.1.2.¶
Finally, the validity circuit in our FLP may have any number of outputs (at least one). The input is said to be valid if each of the outputs is zero. To save bandwidth, we take a random linear combination of the outputs. If each of the outputs is zero, then the reduced output will be zero; but if one of the outputs is non-zero, then the reduced output will be non-zero with high probability.¶
The FLP described in Section 7.3.3 is defined in terms of a
validity circuit Valid
that implements the interface described here.¶
A concrete Valid
defines the following parameters:¶
Parameter | Description |
---|---|
GADGETS
|
A list of gadgets |
GADGET_CALLS
|
Number of times each gadget is called |
MEAS_LEN
|
Length of the measurement |
OUTPUT_LEN
|
Length of the aggregatable output |
JOINT_RAND_LEN
|
Length of the random input |
EVAL_OUTPUT_LEN
|
Length of the circuit output |
Measurement
|
The type of measurement |
AggResult
|
Type of the aggregate result |
field
|
Class object for the field |
Each gadget G
in GADGETS
defines a constant DEGREE
that specifies the
circuit's "arithmetic degree". This is defined to be the degree of the
polynomial that computes it. For example, the Mul
circuit in
Section 7.3.1 is defined by the polynomial Mul(x) = x * x
, which
has degree 2
. Hence, the arithmetic degree of this gadget is 2
.¶
Each gadget also defines a parameter ARITY
that specifies the circuit's arity
(i.e., the number of input wires).¶
Gadgets provide a method to evaluate their circuit on a list of inputs,
eval()
. The inputs can either belong to the validity circuit's field, or the
polynomial ring over that field.¶
A concrete Valid
provides the following methods for encoding a measurement as
an input vector, truncating an input vector to the length of an aggregatable
output, and converting an aggregated output to an aggregate result:¶
valid.encode(measurement: Measurement) -> list[F]
returns a vector of
length MEAS_LEN
representing a measurement.¶
valid.truncate(meas: list[F]) -> list[F]
returns a vector of length
OUTPUT_LEN
representing an aggregatable output.¶
valid.decode(output: list[F], num_measurements: int) -> AggResult
returns an aggregate result.¶
Finally, the following methods are derived for each concrete Valid
:¶
This section specifies an implementation of the Flp
interface (Section 7.1). It
has as a generic parameter a validity circuit Valid
implementing the
interface defined in Section 7.3.2.¶
The parameters are defined in Table 8. The required methods for generating the proof, generating the verifier, and deciding validity are specified in the remaining subsections.¶
In the remainder, we let [n]
denote the set {1, ..., n}
for positive integer
n
. We also define the following constants:¶
Parameter | Value |
---|---|
PROVE_RAND_LEN
|
valid.prove_rand_len() (see Section 7.3.2) |
QUERY_RAND_LEN
|
valid.query_rand_len() (see Section 7.3.2) |
JOINT_RAND_LEN
|
valid.JOINT_RAND_LEN
|
MEAS_LEN
|
valid.MEAS_LEN
|
OUTPUT_LEN
|
valid.OUTPUT_LEN
|
PROOF_LEN
|
valid.proof_len() (see Section 7.3.2) |
VERIFIER_LEN
|
valid.verifier_len() (see Section 7.3.2) |
Measurement
|
valid.Measurement
|
On input of meas
, prove_rand
, and joint_rand
, the proof is computed as
follows:¶
For each i
in [H]
create an empty table wire_i
.¶
Partition the prover randomness prove_rand
into sub-vectors seed_1, ...,
seed_H
where len(seed_i) == L_i
for all i
in [H]
. Let us call these
the "wire seeds" of each gadget.¶
Evaluate Valid
on input of meas
and joint_rand
, recording the inputs
of each gadget in the corresponding table. Specifically, for every i
in
[H]
, set wire_i[j-1,k-1]
to the value on the j
th wire into the k
th
call to gadget G_i
.¶
Compute the "wire polynomials". That is, for every i
in [H]
and j
in
[L_i]
, construct poly_wire_i[j-1]
, the j
th wire polynomial for the
i
th gadget, as follows:¶
Compute the "gadget polynomials". That is, for every i
in [H]
:¶
Let poly_gadget_i = G_i(poly_wire_i[0], ..., poly_wire_i[L_i-1])
. That
is, evaluate the circuit G_i
on the wire polynomials for the i
th
gadget. (Arithmetic is in the ring of polynomials over field
.)¶
The proof is the vector proof = seed_1 + coeff_1 + ... + seed_H + coeff_H
,
where coeff_i
is the vector of coefficients of poly_gadget_i
for each i
in
[H]
.¶
On input of meas
, proof
, query_rand
, and joint_rand
, the verifier message
is generated as follows:¶
For every i
in [H]
create an empty table wire_i
.¶
Partition proof
into the sub-vectors seed_1
, coeff_1
, ..., seed_H
,
coeff_H
defined in Section 7.3.3.1.¶
Evaluate Valid
on input of meas
and joint_rand
, recording the inputs
of each gadget in the corresponding table. This step is similar to the
prover's step (3.) except the verifier does not evaluate the gadgets.
Instead, it computes the output of the k
th call to G_i
by evaluating
poly_gadget_i(alpha_i^k)
. Let out
denote the output of the circuit
evaluation.¶
Next, reduce out
as follows. If EVAL_OUTPUT_LEN > 1
, then consume the
first EVAL_OUTPUT_LEN
elements of query_rand
by letting r, query_rand =
front(1, query_rand)
. Then let
v = r[0]*out[0] + r[1]*out[1] + r[2]*out[2] + ...
.¶
Compute the wire polynomials just as in the prover's step (4.).¶
Compute the tests for well-formedness of the gadget polynomials. That is, for
every i
in [H]
:¶
The verifier message is the vector verifier = [v] + x_1 + [y_1] + ... + x_H +
[y_H]
.¶
On input of vector verifier
, the verifier decides if the measurement is valid
as follows:¶
Parse verifier
into v
, x_1
, y_1
, ..., x_H
, y_H
as defined in
Section 7.3.3.2.¶
Check for well-formedness of the gadget polynomials. For every i
in [H]
:¶
Return True
if v == 0
and False
otherwise.¶
This section specifies instantiations of Prio3 for various measurement types. Each is determined by a field (Section 6.1), a validity circuit (Section 7.3.2), an XOF (Section 6.2). and the number of proofs to generate and verify. Test vectors for each can be found in Appendix "Test Vectors".¶
Parameter | Value |
---|---|
Valid
|
Count(Field64) (this section) |
Field
|
Field64 (Table 3) |
PROOFS
|
1
|
Xof
|
XofTurboShake128 (Section 6.2.1) |
Our first instance of Prio3 is for a simple counter: each measurement is either one or zero and the aggregate result is the sum of the measurements.¶
Its validity circuit, denoted Count
, uses the following degree-2, arity-2
gadget, denoted Mul
:¶
def eval(self, _field: type[F], inp: list[F]) -> F: self.check_gadget_eval(inp) return inp[0] * inp[1]¶
The call to check_gadget_eval()
raises an error if the length of the input is
not equal to the gadget's ARITY
parameter.¶
The Count
validity circuit is defined as¶
def eval( self, meas: list[F], joint_rand: list[F], _num_shares: int) -> list[F]: self.check_valid_eval(meas, joint_rand) squared = self.GADGETS[0].eval(self.field, [meas[0], meas[0]]) return [squared - meas[0]]¶
The measurement is encoded and decoded as a singleton vector in the natural way. The parameters for this circuit are summarized below.¶
Parameter | Value |
---|---|
GADGETS
|
[Mul]
|
GADGET_CALLS
|
[1]
|
MEAS_LEN
|
1
|
OUTPUT_LEN
|
1
|
JOINT_RAND_LEN
|
0
|
EVAL_OUTPUT_LEN
|
1
|
Measurement
|
int in range(2)
|
AggResult
|
int
|
Parameter | Value |
---|---|
Valid
|
Sum(Field64, max_measurement) (this section) |
Field
|
Field64 (Table 3) |
PROOFS
|
1
|
Xof
|
XofTurboShake128 (Section 6.2.1) |
The next instance of Prio3 supports summing of integers in a pre-determined
range. Each measurement is an integer in range(max_measurement+1)
, where
max_measurement
is an associated parameter equal to the largest valid
measurement.¶
The range check is accomplished by encoding the measurement as a bit vector, encoding the measurement plus an offset as a bit vector, then checking that the two encoded integers are consistent. Let¶
bits = max_measurement.bit_length()
, the number of bits needed to encode
the largest valid measurement¶
offset = 2**bits - 1 - max_measurement
¶
The first bit-encoded integer is the measurement itself. Note that only
measurements between 0
and 2**bits - 1
can be encoded this way with
bits
bits. The second bit-encoded integer is the sum of the measurement and
offset
. Observe that this sum can only be encoded this way if it is between 0
and 2**bits - 1
, which implies that the measurement is between -offset
and
max_measurement
.¶
The circuit, denoted Sum
, first checks that each entry of both bit vectors is
a one or a zero. It then decodes both the measurement and the offset
measurement, and subtracts offset
from the latter. It then checks if these
two values are equal. Since both the measurement and the measurement plus
offset
are in the same range of range(2**bits)
, this means that the
measurement itself is between 0
and max_measurement
.¶
The measurement is encoded in 2*bits
field elements as follows:¶
def encode(self, measurement: int) -> list[F]: encoded = [] encoded += self.field.encode_into_bit_vector( measurement, self.bits ) encoded += self.field.encode_into_bit_vector( measurement + self.offset.as_unsigned(), self.bits ) return encoded def truncate(self, meas: list[F]) -> list[F]: return [self.field.decode_from_bit_vector(meas[:self.bits])] def decode(self, output: list[F], _num_measurements: int) -> int: return output[0].as_unsigned()¶
The validity circuit checks that the input consists of ones and zeros. Its
gadget, denoted Range2
, is the degree-2, arity-1 gadget defined as¶
def eval(self, _field: type[F], inp: list[F]) -> F: self.check_gadget_eval(inp) return inp[0] * inp[0] - inp[0]¶
The Sum
validity circuit is defined as¶
def eval( self, meas: list[F], joint_rand: list[F], num_shares: int) -> list[F]: self.check_valid_eval(meas, joint_rand) shares_inv = self.field(num_shares).inv() out = [] for b in meas: out.append(self.GADGETS[0].eval(self.field, [b])) range_check = self.offset * shares_inv + \ self.field.decode_from_bit_vector(meas[:self.bits]) - \ self.field.decode_from_bit_vector(meas[self.bits:]) out.append(range_check) return out¶
Parameter | Value |
---|---|
GADGETS
|
[Range2]
|
GADGET_CALLS
|
[2*bits]
|
MEAS_LEN
|
2*bits
|
OUTPUT_LEN
|
1
|
JOINT_RAND_LEN
|
0
|
EVAL_OUTPUT_LEN
|
2*bits + 1
|
Measurement
|
int in range(max_measurement+1)
|
AggResult
|
int
|
Parameter | Value |
---|---|
Valid
|
SumVec(Field128, length, bits, chunk_lengh) (this section) |
Field
|
Field128 (Table 3) |
PROOFS
|
1
|
Xof
|
XofTurboShake128 (Section 6.2.1) |
This instance of Prio3 supports summing a vector of integers. It has three
parameters, length
, bits
, and chunk_length
. Each measurement is a vector
of positive integers with length equal to the length
parameter. Each element
of the measurement is an integer in the range(2**bits)
. It is RECOMMENDED
to set chunk_length
to an integer near the square root of length * bits
(see Section 7.4.3.1).¶
The validity circuit is denoted SumVec
. Measurements are encoded as a vector
of field elements with length length * bits
. The field elements in the
encoded vector represent all the bits of the measurement vector's elements,
consecutively, in LSB to MSB order:¶
def encode(self, measurement: list[int]) -> list[F]: if len(measurement) != self.length: raise ValueError('incorrect measurement length') encoded = [] for val in measurement: if val not in range(2**self.bits): raise ValueError( 'entry of measurement vector is out of range' ) encoded += self.field.encode_into_bit_vector(val, self.bits) return encoded def truncate(self, meas: list[F]) -> list[F]: truncated = [] for i in range(self.length): truncated.append(self.field.decode_from_bit_vector( meas[i * self.bits: (i + 1) * self.bits] )) return truncated def decode( self, output: list[F], _num_measurements: int) -> list[int]: return [x.as_unsigned() for x in output]¶
This validity circuit uses a ParallelSum
gadget to achieve a smaller proof
size. This optimization for "parallel-sum circuits" is described in
[BBCGGI19], section 4.4. Briefly, for circuits that add up the output of
multiple identical subcircuits, it is possible to achieve smaller proof sizes
(on the order of O(sqrt(MEAS_LEN)) instead of O(MEAS_LEN)) by packaging more
than one such subcircuit into a gadget.¶
The ParallelSum
gadget is parameterized with an arithmetic subcircuit, and a
count
of how many times it evaluates that subcircuit. It takes in a list of
inputs and passes them through to instances of the subcircuit in the same order.
It returns the sum of the subcircuit outputs. Note that only the ParallelSum
gadget itself, and not its subcircuit, participates in the FLP's wire recording
during evaluation, gadget consistency proofs, and proof validation, even though
the subcircuit is provided to ParallelSum
as an implementation of the
Gadget
interface.¶
def eval(self, field: type[F], inp: list[F]) -> F: self.check_gadget_eval(inp) out = field(0) for i in range(self.count): start_index = i * self.subcircuit.ARITY end_index = (i + 1) * self.subcircuit.ARITY out += self.subcircuit.eval( field, inp[start_index:end_index], ) return out¶
The SumVec
validity circuit checks that the encoded measurement consists of
ones and zeros. Rather than use the Range2
gadget on each element, as in the
Sum
validity circuit, it instead uses Mul
subcircuits and "free" constant
multiplication and addition gates to simultaneously evaluate the same range
check polynomial on each element, and multiply by a constant. One of the two
Mul
subcircuit inputs is equal to a measurement element multiplied by a power
of one of the elements of the joint randomness vector, and the other is equal
to the same measurement element minus one. These Mul
subcircuits are
evaluated by a ParallelSum
gadget, and the results are added up both within
the ParallelSum
gadget and after it.¶
def eval( self, meas: list[F], joint_rand: list[F], num_shares: int) -> list[F]: self.check_valid_eval(meas, joint_rand) out = self.field(0) shares_inv = self.field(num_shares).inv() for i in range(self.GADGET_CALLS[0]): r = joint_rand[i] r_power = r inputs: list[Optional[F]] inputs = [None] * (2 * self.chunk_length) for j in range(self.chunk_length): index = i * self.chunk_length + j if index < len(meas): meas_elem = meas[index] else: meas_elem = self.field(0) inputs[j * 2] = r_power * meas_elem inputs[j * 2 + 1] = meas_elem - shares_inv r_power *= r out += self.GADGETS[0].eval( self.field, cast(list[F], inputs), ) return [out]¶
Parameter | Value |
---|---|
GADGETS
|
[ParallelSum(Mul(), chunk_length)]
|
GADGET_CALLS
|
[(length * bits + chunk_length - 1) // chunk_length]
|
MEAS_LEN
|
length * bits
|
OUTPUT_LEN
|
length
|
JOINT_RAND_LEN
|
GADGET_CALLS[0]
|
EVAL_OUTPUT_LEN
|
1
|
Measurement
|
list[int] , each element in range(2**bits)
|
AggResult
|
list[int]
|
ParallelSum
chunk length
The chunk_length
parameter provides a trade-off between the arity of the
ParallelSum
gadget and the number of times the gadget is called. The proof
length is asymptotically minimized when the chunk length is near the square root
of the length of the measurement. However, the relationship between VDAF
parameters and proof length is complicated, involving two forms of rounding (the
circuit pads the inputs to its last ParallelSum
gadget call, up to the chunk
length, and proof system rounds the degree of wire polynomials -- determined by
the number of times a gadget is called -- up to the next power of two).
Therefore, the optimal choice of chunk_length
for a concrete measurement size
will vary, and must be found through trial and error. Setting chunk_length
equal to the square root of the appropriate measurement length will result in
proofs up to 50% larger than the optimal proof size.¶
Parameter | Value |
---|---|
Valid
|
Histogram(Field128, length, chunk_lengh) (this section) |
Field
|
Field128 (Table 3) |
PROOFS
|
1
|
Xof
|
XofTurboShake128 (Section 6.2.1) |
This instance of Prio3 allows for estimating the distribution of some quantity
by computing a simple histogram. Each measurement increments one histogram
bucket, out of a set of fixed buckets. (Bucket indexing begins at 0
.) For
example, the buckets might quantize the real numbers, and each measurement would
report the bucket that the corresponding client's real-numbered value falls
into. The aggregate result counts the number of measurements in each bucket.¶
The validity circuit is denoted Histogram
. It has two parameters, length
,
the number of histogram buckets, and chunk_length
, which is used by by a
circuit optimization described below. It is RECOMMENDED to set chunk_length
to an integer near the square root of length
(see
Section 7.4.3.1).¶
The measurement is encoded as a one-hot vector representing the bucket into which the measurement falls:¶
def encode(self, measurement: int) -> list[F]: encoded = [self.field(0)] * self.length encoded[measurement] = self.field(1) return encoded def truncate(self, meas: list[F]) -> list[F]: return meas def decode( self, output: list[F], _num_measurements: int) -> list[int]: return [bucket_count.as_unsigned() for bucket_count in output]¶
The Histogram
validity circuit checks for one-hotness in two steps, by
checking that the encoded measurement consists of ones and zeros, and by
checking that the sum of all elements in the encoded measurement is equal to
one. The individual checks constitute the output of the circuit.¶
As in the SumVec
validity circuit (Section 7.4.3), the first part of the
validity circuit uses the ParallelSum
gadget to perform range checks while
achieving a smaller proof size. The ParallelSum
gadget uses Mul
subcircuits
to evaluate a range check polynomial on each element, and includes an additional
constant multiplication. One of the two Mul
subcircuit inputs is equal to a
measurement element multiplied by a power of an element of the joint randomness
vector, and the other is equal to the same measurement element minus one. The
results are added up both within the ParallelSum
gadget and after it.¶
def eval( self, meas: list[F], joint_rand: list[F], num_shares: int) -> list[F]: self.check_valid_eval(meas, joint_rand) # Check that each bucket is one or zero. range_check = self.field(0) shares_inv = self.field(num_shares).inv() for i in range(self.GADGET_CALLS[0]): r = joint_rand[i] r_power = r inputs: list[Optional[F]] inputs = [None] * (2 * self.chunk_length) for j in range(self.chunk_length): index = i * self.chunk_length + j if index < len(meas): meas_elem = meas[index] else: meas_elem = self.field(0) inputs[j * 2] = r_power * meas_elem inputs[j * 2 + 1] = meas_elem - shares_inv r_power *= r range_check += self.GADGETS[0].eval( self.field, cast(list[F], inputs), ) # Check that the buckets sum to 1. sum_check = -shares_inv for b in meas: sum_check += b return [range_check, sum_check]¶
Note that this circuit depends on the number of shares into which the measurement is sharded. This is provided to the FLP by Prio3.¶
Parameter | Value |
---|---|
GADGETS
|
[ParallelSum(Mul(), chunk_length)]
|
GADGET_CALLS
|
[(length + chunk_length - 1) // chunk_length]
|
MEAS_LEN
|
length
|
OUTPUT_LEN
|
length
|
JOINT_RAND_LEN
|
GADGET_CALLS[0]
|
EVAL_OUTPUT_LEN
|
2
|
Measurement
|
int
|
AggResult
|
list[int]
|
Parameter | Value |
---|---|
Valid
|
MultihotCountVec(Field128, length, max_weight, chunk_lengh) (this section) |
Field
|
Field128 (Table 3) |
PROOFS
|
1
|
Xof
|
XofTurboShake128 (Section 6.2.1) |
For this instance of Prio3, each measurement is a vector of ones and zeros, where the number of ones is bounded. This provides a functionality similar to Prio3Histogram except that more than one entry may be non-zero. This allows Prio3MultihotCountVec to be composed with a randomized response mechanism, like [EPK14], for providing differential privacy. (For example, each Client would set each entry to one with some small probability.)¶
Prio3MultihotCountVec uses XofTurboShake128 (Section 6.2.1) as its XOF.
Its validity circuit is denoted MultihotCountVec
. It has three parameters:
length
, the number of of entries in the count vector; max_weight
, the
maximum number of non-zero entries (i.e., the weight must be at most
max_weight
); and chunk_length
, used the same way as in Section 7.4.3 and
Section 7.4.4.¶
Validation works as follows. Let¶
The Client reports the weight of the count vector by adding offset
to it and
bit-encoding the result. Observe that only a weight of at most max_weight
can
be encoded with bits_for_weight
bits.¶
The verifier checks that each entry of the encoded measurement is a bit (i.e.,
either one or zero). It then decodes the reported weight and subtracts it from
offset + sum(count_vec)
, where count_vec
is the count vector. The result is
zero if and only if the reported weight is equal to the true weight.¶
Encoding, truncation, and decoding are defined as follows:¶
def encode(self, measurement: list[int]) -> list[F]: if len(measurement) != self.length: raise ValueError('invalid Client measurement length') # The first part is the vector of counters. count_vec = list(map(self.field, measurement)) # The second part is the reported weight. weight_reported = sum(count_vec, self.field(0)) encoded = [] encoded += count_vec encoded += self.field.encode_into_bit_vector( (self.offset + weight_reported).as_unsigned(), self.bits_for_weight) return encoded def truncate(self, meas: list[F]) -> list[F]: return meas[:self.length] def decode( self, output: list[F], _num_measurements: int) -> list[int]: return [bucket_count.as_unsigned() for bucket_count in output]¶
Circuit evaluation is defined as follows:¶
def eval( self, meas: list[F], joint_rand: list[F], num_shares: int) -> list[F]: self.check_valid_eval(meas, joint_rand) # Check that each entry in the input vector is one or zero. range_check = self.field(0) shares_inv = self.field(num_shares).inv() for i in range(self.GADGET_CALLS[0]): r = joint_rand[i] r_power = r inputs: list[Optional[F]] inputs = [None] * (2 * self.chunk_length) for j in range(self.chunk_length): index = i * self.chunk_length + j if index < len(meas): meas_elem = meas[index] else: meas_elem = self.field(0) inputs[j * 2] = r_power * meas_elem inputs[j * 2 + 1] = meas_elem - shares_inv r_power *= r range_check += self.GADGETS[0].eval( self.field, cast(list[F], inputs), ) # Check that the weight `offset` plus the sum of the counters # is equal to the value claimed by the Client. count_vec = meas[:self.length] weight = sum(count_vec, self.field(0)) weight_reported = \ self.field.decode_from_bit_vector(meas[self.length:]) weight_check = self.offset*shares_inv + weight - \ weight_reported return [range_check, weight_check]¶
Parameter | Value |
---|---|
GADGETS
|
[ParallelSum(Mul(), chunk_length)]
|
GADGET_CALLS
|
[(length + bits_for_weight + chunk_length - 1) // chunk_length]
|
MEAS_LEN
|
length + bits_for_weight
|
OUTPUT_LEN
|
length
|
JOINT_RAND_LEN
|
GADGET_CALLS[0]
|
EVAL_OUTPUT_LEN
|
2
|
Measurement
|
list[int]
|
AggResult
|
list[int]
|
This section specifies Poplar1, a VDAF for the following task. Each Client
holds a bit-string of length BITS
and the Aggregators hold a sequence of
L
-bit strings, where L <= BITS
. We will refer to the latter as the set of
"candidate prefixes". The Aggregators' goal is to count how many measurements
are prefixed by each candidate prefix.¶
This functionality is the core component of the Poplar protocol [BBCGGI21], which was designed to compute the heavy hitters over a set of input strings. At a high level, the protocol works as follows.¶
Each Client splits its string into input shares and sends one share to each Aggregator.¶
The Aggregators agree on an initial set of candidate prefixes, say 0
and
1
.¶
The Aggregators evaluate the VDAF on each set of input shares and aggregate the recovered output shares. The aggregation parameter is the set of candidate prefixes.¶
The Aggregators send their aggregate shares to the Collector, who combines them to recover the counts of each candidate prefix.¶
Let H
denote the set of prefixes that occurred at least t
times. If the
prefixes all have length BITS
, then H
is the set of t
-heavy-hitters.
Otherwise compute the next set of candidate prefixes, e.g., for each p
in
H
, add p || 0
and p || 1
to the set. Repeat step 3 with the new set of
candidate prefixes.¶
Poplar1 is constructed from an "Incremental Distributed Point Function (IDPF)", a primitive described by [BBCGGI21] that generalizes the notion of a Distributed Point Function (DPF) [GI14]. Briefly, a DPF is used to distribute the computation of a "point function", a function that evaluates to zero on every input except at a programmable "point". The computation is distributed in such a way that no one party knows either the point or what it evaluates to.¶
An IDPF generalizes this "point" to a path on a full binary tree from the root to one of the leaves. It is evaluated on an "index" representing a unique node of the tree. If the node is on the programmed path, then the function evaluates to a non-zero value; otherwise it evaluates to zero. This structure allows an IDPF to provide the functionality required for the above protocol: to compute the hit count for an index, just evaluate each set of IDPF shares at that index and add up the results.¶
Consider the sub-tree constructed from a set of input strings and a target
threshold t
by including all indices that prefix at least t
of the input
strings. We shall refer to this structure as the "prefix tree" for the batch of
inputs and target threshold. To compute the t
-heavy hitters for a set of
inputs, the Aggregators and Collector first compute the prefix tree, then
extract the heavy hitters from the leaves of this tree. (Note that the prefix
tree may leak more information about the set than the heavy hitters themselves;
see Section 9.4.1 for details.)¶
Poplar1 composes an IDPF with the arithmetic sketch of [BBCGGI21], Section 4.2. (The paper calls this a "secure sketch", but the underlying technique was later generalized in [BBCGGI23], where it is called "arithmetic sketching".) This protocol ensures that evaluating a set of input shares on a unique set of candidate prefixes results in shares of a "one-hot" vector, i.e., a vector that is zero everywhere except for at most one element, which is equal to one.¶
The remainder of this section is structured as follows. IDPFs are defined in Section 8.1; a concrete instantiation is given Section 8.3. The Poplar1 VDAF is defined in Section 8.2 in terms of a generic IDPF. Finally, a concrete instantiation of Poplar1 is specified in Section 8.4; test vectors can be found in Appendix "Test Vectors".¶
An IDPF is defined over a domain of size 2^BITS
, where BITS
is a constant.
Indices into the IDPF tree are bit strings. (In Poplar1, each Client's bit
string is an index; see Section 8.1.1 for details.) The Client
specifies an index alpha
and a vector of values beta
, one for each "level"
L
in range(BITS)
. The key generation algorithm generates one IDPF "key" for
each Aggregator. When evaluated at level L
and index prefix
, each IDPF key
returns an additive share of beta[L]
if prefix
is the L
-bit prefix of
alpha
and shares of zero otherwise.¶
Each of the programmed points beta
is a vector of elements of some finite
field. We distinguish two types of fields: one for inner nodes (denoted
FieldInner
), and one for leaf nodes (FieldLeaf
). (Our
instantiation of Poplar1 (Section 8.4) will use a much larger field for
leaf nodes than for inner nodes. This is to ensure the IDPF is "extractable" as
defined in [BBCGGI21], Definition 1.)¶
A concrete IDPF defines the types and constants enumerated in Table 19.
In the remainder we write Output
as shorthand for the type
list[list[FieldInner]] | list[list[FieldLeaf]]
. (This type
denotes either a vector of inner node field elements or leaf node field
elements.) The scheme is comprised of the following algorithms:¶
idpf.gen(alpha: tuple[bool, ...], beta_inner: list[list[FieldInner]], beta_leaf:
list[FieldLeaf], ctx: bytes, nonce: bytes, rand: bytes) -> tuple[PublicShare,
list[bytes]]
is the randomized IDPF-key generation algorithm. Its inputs are
the index alpha
, the values beta
, an application context string, and a
nonce string.¶
The output is a public part (of type PublicShare
) that is sent to all
Aggregators and a vector of private IDPF keys, one for each aggregator. The
binder string is used to derive the key in the underlying XofFixedKeyAes128
XOF that is used for expanding seeds at each level.¶
Pre-conditions:¶
alpha
MUST have length BITS
.¶
beta_inner
MUST have length BITS - 1
.¶
beta_inner[level]
MUST have length VALUE_LEN
for each level
in
range(BITS - 1)
.¶
beta_leaf
MUST have length VALUE_LEN
.¶
rand
MUST be generated by a CSPRNG and have length RAND_SIZE
.¶
nonce
MUST be of length Idpf.NONCE_SIZE
and chosen uniformly at
random by the Client (see Section 9.2).¶
idpf.eval(agg_id: int, public_share: PublicShare, key: bytes, level: int,
prefixes: Sequence[tuple[bool, ...]], ctx: bytes, nonce: bytes) -> Output
is
the deterministic, stateless IDPF-key evaluation algorithm run by each
Aggregator. Its inputs are the Aggregator's unique identifier, the public
share distributed to all of the Aggregators, the Aggregator's IDPF key, the
"level" at which to evaluate the IDPF, the sequence of candidate prefixes, an
application context string, and a nonce string. It returns the share of the
value corresponding to each candidate prefix.¶
The output type (i.e., Output
) depends on the value of level
: if level <
BITS-1
, the output is the value for an inner node, which has type
list[list[FieldInner]]
; otherwise, if level == BITS-1
, then the output is
the value for a leaf node, which has type list[list[FieldLeaf]]
.¶
Pre-conditions:¶
In addition, the following method is derived for each concrete Idpf
:¶
def current_field( self, level: int) -> type[FieldInner] | type[FieldLeaf]: if level < self.BITS - 1: return self.field_inner return self.field_leaf¶
Finally, an implementation note. The interface for IDPFs specified here is stateless, in the sense that there is no state carried between IDPF evaluations. This is to align the IDPF syntax with the VDAF abstraction boundary, which does not include shared state across VDAF evaluations. In practice, of course, it will often be beneficial to expose a stateful API for IDPFs and carry the state across evaluations. See Section 8.3 for details.¶
Parameter | Description |
---|---|
SHARES | Number of IDPF keys output by IDPF-key generator |
BITS | Length in bits of each input string |
VALUE_LEN | Number of field elements of each output value |
RAND_SIZE | Size of the random string consumed by the IDPF-key generator. Equal to twice the XOF's seed size. |
NONCE_SIZE | Size of the randon nonce generated by the Client. |
KEY_SIZE | Size in bytes of each IDPF key |
FieldInner | Implementation of Field (Section 6.1) used for values of inner nodes |
FieldLeaf | Implementation of Field used for values of leaf nodes |
PublicShare | Type of public share for this IDPF |
Output | Alias of list[list[FieldInner]] | list[list[FieldLeaf]]
|
FieldVec | Alias of list[FieldInner] | list[FieldLeaf]
|
How data are represented as IDPF indices is up to the application. When the inputs are fixed-length byte strings, the most natural choice of representation is as a bit string formed from all the bits of the byte string, first ordered by byte position, then ordered from most significant bit to least significant bit within each byte. This ensures that, when a byte string is a prefix of another, so too is its corresponding index. (Index prefixes are defined in Section 8.1.) For example,¶
Byte string: 01 02 Bit string: 00000001 00000010¶
is a prefix of¶
Byte string: 01 02 03 Bit string: 00000001 00000010 00000011¶
Additionally, lexicographic ordering is preserved by this mapping from a byte string to a bit string.¶
When the inputs are variable length, it is necessary to pad each input to some
fixed length. Further, the padding scheme must be non-ambiguous. For example,
each input could be padded with b"\x01"
followed by as many b"\x00"
bytes
as needed.¶
This section specifies Poplar1
, an implementation of the Vdaf
interface
(Section 5). It is defined in terms of any Idpf
(Section 8.1) for which
SHARES == 2
and VALUE_LEN == 2
and an implementation of Xof
(Section 6.2). The associated constants and types required by the Vdaf
interface
are defined in Table 20. The methods required for sharding,
preparation, aggregation, and unsharding are described in the remaining
subsections. These methods make use of constants defined in Table 21.¶
Parameter | Value |
---|---|
VERIFY_KEY_SIZE
|
Xof.SEED_SIZE
|
RAND_SIZE
|
Xof.SEED_SIZE * 3 + Idpf.RAND_SIZE
|
NONCE_SIZE
|
16
|
ROUNDS
|
2
|
SHARES
|
2
|
Measurement
|
tuple[bool, ...]
|
AggParam
|
tuple[int, Sequence[tuple[bool, ...]]]
|
PublicShare
|
same as the IDPF |
InputShare
|
tuple[bytes, bytes, list[FieldInner], list[FieldLeaf]]
|
OutShare
|
FieldVec
|
AggShare
|
FieldVec
|
AggResult
|
list[int]
|
PrepState
|
tuple[bytes, int, FieldVec]
|
PrepShare
|
FieldVec
|
PrepMessage
|
Optional[FieldVec]
|
Variable | Value |
---|---|
USAGE_SHARD_RAND: int | 1 |
USAGE_CORR_INNER: int | 2 |
USAGE_CORR_LEAF: int | 3 |
USAGE_VERIFY_RAND: int | 4 |
The aggregation parameter encodes a sequence of candidate prefixes. When an
Aggregator receives an input share from the Client, it begins by evaluating its
IDPF share on each candidate prefix, recovering a data_share
and auth_share
for each. The Aggregators use these and the correlation shares provided by the
Client to verify that the sequence of data_share
values are additive shares of
a one-hot vector.¶
Aggregators MUST ensure the candidate prefixes are all unique and appear in
lexicographic order. (This is enforced in the definition of prep_init()
below.) Uniqueness is necessary to ensure the refined measurement (i.e., the sum
of the output shares) is in fact a one-hot vector. Otherwise, sketch
verification might fail, causing the Aggregators to erroneously reject a report
that is actually valid. Note that enforcing the order is not strictly necessary,
but this does allow uniqueness to be determined more efficiently.¶
Aggregation parameters are valid for a given input share if no aggregation parameter with the same level has been used with the same input share before. The whole preparation phase MUST NOT be run more than once for a given combination of input share and level. This function checks that levels are increasing between calls, and also enforces that the prefixes at each level are suffixes of the previous level's prefixes.¶
Aggregation involves simply adding up the output shares.¶
Finally, the Collector unshards the aggregate result by adding up the aggregate shares.¶
This section defines serialization formats for messages exchanged over the
network while executing Poplar1
. It is RECOMMENDED that implementations
provide serialization methods for them.¶
Message structures are defined following Section 3 of [RFC8446]). In the
remainder we use Fi
as an alias for poplar1.idpf.field_inner.ENCODED_SIZE
,
Fl
as an alias for poplar1.idpf.field_leaf.ENCODED_SIZE
, and B
as an
alias for poplar1.idpf.BITS
.¶
Elements of the inner field are encoded in little-endian byte order (as defined in Section 6.1) and are represented as follows:¶
opaque Poplar1FieldInner[Fi];¶
Likewise, elements of the leaf field are encoded in little-endian byte order (as defined in Section 6.1) and are represented as follows:¶
opaque Poplar1FieldLeaf[Fl];¶
Likewise, the structure of the prep message for Poplar1 depends on the sketching round and field. For the first round and inner field:¶
struct { Poplar1FieldInner[Fi * 3]; } Poplar1PrepMessageRoundOneInner;¶
For the first round and leaf field:¶
struct { Poplar1FieldLeaf sketch[Fl * 3]; } Poplar1PrepMessageRoundOneLeaf;¶
Note that these messages have the same structures as the prep shares for the first round.¶
The second-round prep message is the empty string. This is because the sketch shares are expected to sum to a particular value if the output shares are valid; we represent a successful preparation with the empty string and otherwise return an error.¶
The aggregation parameter is encoded as follows:¶
struct { uint16_t level; uint32_t num_prefixes; opaque encoded_prefixes[prefixes_len]; } Poplar1AggParam;¶
The fields in this struct are: level
, the level of the IDPF tree of each
prefixes; num_prefixes
, the number of prefixes to evaluate; and
encoded_prefixes
, the sequence of prefixes encoded into a byte string of
length prefixes_len
. The prefixes are encoded with the following procedure:¶
prefixes_len = ((level + 1) + 7) // 8 * len(prefixes) encoded_prefixes = bytearray() for prefix in prefixes: for chunk in itertools.batched(prefix, 8): byte_out = 0 for (bit_position, bit) in enumerate(chunk): byte_out |= bit << (7 - bit_position) encoded_prefixes.append(byte_out)¶
Decoding involves the following procedure:¶
prefixes = [] bytes_per_prefix = ((level + 1) + 7) // 8 for chunk in itertools.batched(encoded_prefixes, bytes_per_prefix): prefix = [] for i in range(level + 1): byte_index = i // 8 bit_offset = 7 - (i % 8) bit = (chunk[byte_index] >> bit_offset) & 1 != 0 prefix.append(bit) prefixes.append(tuple(prefix))¶
Implementation note: the aggregation parameter includes the level of the IDPF tree and the sequence of indices to evaluate. For implementations that perform per-report caching across executions of the VDAF, this may be more information than is strictly needed. In particular, it may be sufficient to convey which indices from the previous execution will have their children included in the next. This would help reduce communication overhead.¶
In this section we specify a concrete IDPF suitable for instantiating Poplar1. The scheme gets its name from the name of the protocol of [BBCGGI21].¶
The constant and type definitions required by the Idpf
interface are given in
Table 22.¶
Our IDPF requires an XOF for deriving the output shares, as well as a variety of other artifacts used internally. For performance reasons, we instantiate this object using XofFixedKeyAes128 (Section 6.2.2) wherever possible. See Section 9.5 for more information.¶
Parameter | Value |
---|---|
SHARES |
2
|
BITS | any positive integer |
VALUE_LEN | any positive integer |
KEY_SIZE |
Xof.SEED_SIZE
|
FieldInner |
Field64 (Table 3) |
FieldLeaf |
Field255 (Table 3) |
TODO Describe the construction in prose, beginning with a gentle introduction to the high level idea.¶
The description of the IDPF-key generation algorithm makes use of auxiliary
functions extend()
and convert()
defined in
Section 8.3.3. In the following, we let Field2
denote
the field GF(2)
.¶
TODO Describe in prose how IDPF-key evaluation algorithm works.¶
The description of the IDPF-evaluation algorithm makes use of auxiliary
functions extend()
and convert()
defined in
Section 8.3.3.¶
By default, Poplar1 is instantiated with the IDPF in Section 8.3 (VALUE_LEN
== 2
) and XofTurboShake128 (Section 6.2.1). This VDAF is suitable for
any positive value of BITS
. Test vectors can be found in Appendix "Test Vectors".¶
VDAFs (Section 5) have two essential security goals:¶
Privacy: an attacker that controls the Collector and a subset of Clients and Aggregators learns nothing about the measurements of honest Clients beyond what it can deduce from the aggregate result. We assume the attacker controls the entire network except for channels between honest Clients and honest Aggregators. In particular, it cannot forge or prevent transmission of messages on these channels.¶
Robustness: an attacker that controls a subset of Clients cannot cause the Collector to compute anything other than the aggregate of the measurements of honest Clients. We assume the attacker eavesdrops on the network but does not control transmission of messages between honest parties.¶
Formal definitions of privacy and robustness can be found in [DPRS23]. A VDAF is the core cryptographic primitive of a protocol that achieves the above privacy and robustness goals. It is not sufficient on its own, however. The application will need to assure a few security properties, for example:¶
Securely distributing the long-lived parameters, in particular the verification key.¶
Establishing secure channels:¶
Enforcing the non-collusion properties required of the specific VDAF in use.¶
In such an environment, a VDAF provides the high-level privacy property described above: the Collector learns only the aggregate measurement, and nothing about individual measurements aside from what can be inferred from the aggregate result. The Aggregators learn neither individual measurements nor the aggregate result. The Collector is assured that the aggregate statistic accurately reflects the inputs as long as the Aggregators correctly executed their role in the VDAF.¶
On their own, VDAFs do not provide:¶
Mitigation of Sybil attacks [Dou02]. In this attack, the adversary observes a subset of input shares transmitted by a Client it is interested in. It allows the input shares to be processed, but corrupts and picks bogus measurements for the remaining Clients. Applications can guard against these risks by adding additional controls on report submission, such as Client authentication and rate limits.¶
Differential privacy [Dwo06]. Depending on the distribution of the measurements, the aggregate result itself can still leak a significant amount of information about an individual measurement or the person that generated it.¶
Robustness in the presence of a malicious Aggregator. An Aggregator can, without detection, manipulate the aggregate result by modifying its own aggregate share.¶
Guaranteed output delivery [GSZ20]. An attacker that controls transmission of messages between honest parties can prevent computation of the aggregate result by dropping messages.¶
The Aggregators are responsible for exchanging the verification key in advance of executing the VDAF. Any procedure is acceptable as long as the following conditions are met:¶
To ensure robustness of the computation, the Aggregators MUST NOT reveal the verification key to the Clients. Otherwise, a malicious Client might be able to exploit knowledge of this key to craft an invalid report that would be accepted by the Aggregators.¶
To ensure privacy of the measurements, the Aggregators MUST commit to the verification key prior to processing reports generated by Clients. Otherwise, a malicious Aggregator may be able to craft a verification key that, for a given report, causes an honest Aggregator to leak information about the measurement during preparation.¶
Meeting these conditions is required in order to leverage security analysis in the framework of [DPRS23]. Their definition of robustness allows the attacker, playing the role of a cohort of malicious Clients, to submit arbitrary reports to the Aggregators and eavesdrop on their communications as they process them. Security in this model is achievable as long as the verification key is kept secret from the attacker.¶
The privacy definition of [DPRS23] considers an active attacker that controls the network and a subset of Aggregators; in addition, the attacker is allowed to choose the verification key used by each honest Aggregator over the course of the experiment. Security is achievable in this model as long as the key is picked at the start of the experiment, prior to any reports being generated. (The model also requires nonces to be generated at random; see Section 9.2 below.)¶
Meeting these requirements is relatively straightforward. For example, the Aggregators may designate one of their peers to generate the verification key and distribute it to the others. To assure Clients of key commitment, the Clients and (honest) Aggregators could bind reports to a shared context string derived from the key. For instance, the "task ID" of DAP [DAP] could be set to the hash of the verification key; then as long as honest Aggregators only consume reports for the task indicated by the Client, forging a new key after the fact would reduce to finding collisions in the underlying hash function. (Keeping the key secret from the Clients would require the hash function to be one-way.) However, since rotating the key implies rotating the task ID, this scheme would not allow key rotation over the lifetime of a task.¶
The sharding and preparation steps of VDAF execution depend on a nonce associated with the Client's report. To ensure privacy of the underlying measurement, the Client MUST generate this nonce using a CSPRNG. This is required in order to leverage security analysis for the privacy definition of [DPRS23], which assumes the nonce is chosen at random prior to generating the report.¶
Other security considerations may require the nonce to be non-repeating. For example, to achieve differential privacy it is necessary to avoid "over exposing" a report by including it too many times in a single batch or across multiple batches. It is RECOMMENDED that the nonce generated by the Client be used by the Aggregators for replay protection.¶
As described in Section 4.3 and Section 5.3 respectively, DAFs and VDAFs may impose restrictions on the re-use of input shares. This is to ensure that correlated randomness provided by the Client through the input share is not used more than once, which might compromise confidentiality of the Client's measurements.¶
Protocols that make use of VDAFs therefore MUST call vdaf.is_valid
on the set of all aggregation parameters used for a Client's input share, and
only proceed with the preparation and aggregation phases if that function call
returns True
.¶
Aggregating a batch of reports multiple times, each time with a different aggregation parameter, could result in information leakage beyond what is used by the application.¶
For example, when Poplar1 is used for heavy hitters, the Aggregators learn not only the heavy hitters themselves, but also the prefix tree (as defined in Section 8) computed along the way. Indeed, this leakage is inherent to any construction that uses an IDPF (Section 8.1) in the same way. Depending on the distribution of the measurements, the prefix tree can leak a significant amount of information about unpopular inputs. For instance, it is possible (though perhaps unlikely) for a large set of non-heavy-hitter values to share a common prefix, which would be leaked by a prefix tree with a sufficiently small threshold.¶
A malicious adversary controlling the Collector and one of the Aggregators can further turn arbitrary non-heavy prefixes into heavy ones by tampering with the IDPF output at any position. While our construction ensures that the nodes evaluated at one level are children of the nodes evaluated at the previous level, this still may allow an adversary to discover individual non-heavy strings.¶
The only practical, general-purpose approach to mitigating these leakages is via differential privacy, which is RECOMMENDED for all protocols using Poplar1 for heavy-hitter type applications.¶
The arithmetic sketch described in Section 8 is used by the Aggregators to check
that the shares of the vector obtained by evaluating a Client's IDPF at a
sequence of candidate prefixes has at most one non-zero value, and that the
non-zero value is 1
. Depending on how the values are used, the arithmetic sketch
on its own may not be sufficient for robustness of the application. In
particular, a malicious Client may attempt to influence the computation by
choosing an IDPF that evaluates to 1
at more than one node at a given
level of the tree.¶
This issue can be mitigated by using an IDPF that is extractable as defined in in Appendix D of [BBCGGI21]. Extractability ensures that, for a particular level of the tree, it is infeasible for an attacker to control values of the IDPF such that it takes on chosen non-zero values at more than one node. (It can practically only achieve the zero function, a point function, or a pseudorandom function.)¶
The IDPF specified in Section 8.1 only guarantees extractability at the last level of the tree. (This is by virtue of using a larger field for the leaves than for inner nodes and using an XOF to derive leaves that is safe to model as a random oracle (see Section 9.5).) For intermediate levels, it is feasible for a client to produce IDPF shares with two controlled non-zero nodes.¶
This is not an issue for running heavy hitters, since (1) each node in the prefix tree is a child of a previously traversed node, (2) the arithmetic sketch would detect double voting at every level of the prefix tree, and (3) the IDPF is extractable at the last level of the tree. However, the lack of extractability at intermediate levels may result in attacks on the robustness of certain applications.¶
Thus applications SHOULD NOT use prefix counts for intermediate levels for any purpose beyond the heavy-hitters tree traversal.¶
As described in Section 6.2, our constructions rely on eXtendable Output Functions (XOFs). In the security analyses of our protocols, these are usually modeled as random oracles. XofTurboShake128 is designed to be indifferentiable from a random oracle [MRH04], making it a suitable choice for most situations.¶
The one exception is the IDPF of Section 8.3. Here, a random oracle is not
needed to prove privacy, since the analysis of [BBCGGI21], Proposition 1, only
requires a Pseudorandom Generator (PRG). As observed in [GKWY20], a PRG can be
instantiated from a correlation-robust hash function H
. Informally,
correlation robustness requires that for a random r
, H(xor(r, x))
is
computationally indistinguishable from a random function of x
. A PRG can
therefore be constructed as¶
PRG(r) = H(xor(r, 1)) || H(xor(r, 2)) || ...¶
since each individual hash function evaluation is indistinguishable from a random function.¶
Our construction at Section 6.2.2 implements a correlation-robust hash function using fixed-key AES. For security, it assumes that AES with a fixed key can be modeled as a random permutation [GKWY20]. Additionally, we use a different AES key for every client, which in the ideal cipher model leads to better concrete security [GKWWY20].¶
We note that for robustness, the analysis of [BBCGGI21] still assumes a random oracle to make the IDPF extractable. We therefore use XofTurboShake128 instead for the last level of the tree. It is important that XofTurboShake128 supports 16 byte seeds, as this is the seed size for the inner levels.¶
While XofFixedKeyAes128 has been shown to be differentiable from a random oracle [GKWWY20], there are no known attacks exploiting this difference. We also stress that even if the IDPF is not extractable, Poplar1 guarantees that every client can contribute to at most one prefix among the ones being evaluated by the helpers.¶
Prio3 and other systems built from FLPs (Section 7.3 in particular) may benefit from choosing a field size that is as small as possible. Generally speaking, a smaller field results in lower communication and storage costs. Care must be taken, however, since a smaller field also results in degraded (or even vacuous) robustness.¶
Different variants of Prio3 (Section 7) use different field sizes: Prio3Count and Prio3Sum use Field64; but Prio3SumVec, Prio3Histogram, and Prio3MultihotCountVec all use Field128, a field that is twice as large as Field64. This is due to the use of joint randomness (Section 7.1) in the latter variants. Joint randomness allows for more flexible circuit design (see Section 7.3.1.1), but opens up Prio3 to offline attacks in which the attacker searches for input shares for an invalid measurement that derive joint randomness that causes the circuit to accept. Choosing a large enough field ensures this computation is too expensive to be feasible. (See [DPRS23], Theorem 1.) Note that privacy is not susceptible to such attacks.¶
Another way to mitigate this issue (or improve robustness in general) is to
generate and verify multiple, independent proofs. (See Section 7.1.2.) For
Prio3, the PROOFS
parameter controls the number of proofs (at least one) that
are generated and verified. In general the soundness error of the FLP is given
by the following formula:¶
(circuit_soundness + flp_soundness) ** PROOFS¶
where:¶
circuit_soundness
is the soundness of the validity circuit
(Section 7.3.2)¶
flp_soundness
is the base soundness of the proof system ([BBCGGI19],
Theorem 4.3)¶
For circuits involving joint randomness, we aim for the soundness error to be
close to 2^-128
in order to mitigate offline attacks. Such circuits MUST use
Field128 with at least one proof or Field64 with at least three proofs.
Depending on the circuit, Field64 with two proofs might have significantly
lower soundness than Field128 with one proof.¶
We stress that weak parameters (too small a field, too few proofs, or both) can be exploited to attack any aggregation task using those parameters. To mitigate offline attacks, it is necessary to disable all tasks that use the weak parameters.¶
Two Aggregators are required for privacy in our threat model, but some (V)DAFs, including Prio3 (Section 7), allow for any number of Aggregators, only one of which needs to be trusted in order for the computation to be private. To hedge against corruptions that happen during the course of the attack, deployments may consider involving more than two Aggregators as described for example in Section 5.9. Note however that some schemes are not compatible with this mode of operation, such as Poplar1.¶
Prio3 and Poplar1 are designed to resist some attacks that fall outside the main threat model for VDAFs.¶
Broadly speaking, domain separation is used to prevent cross protocol attacks, in which data from evaluation of one VDAF translates to an attack against another. For example:¶
Weak entropy sources: the VDAF algorithm ID is bound to each XOF invocation, thereby ensuring the outputs are different between VDAF invocations, even if the underlying randomness is the same. For example, two different instances of Prio3 would compute different measurement shares.¶
Weak parameters: Prio3 variants that require joint randomness are subject to offline attacks against robustness. These attacks are feasible if the field size or number of proofs is sufficiently small. (See Section 9.6.) The joint randomness derivation is bound to both the field (via the algorithm ID) and the number of proofs, thereby ensuring that joint randomness derived for weak parameters is not reused for stronger parameters. In addition, the joint randomness is bound to the application context, meaning any work the attacker does to attack some application is not useful for other applications that use the same parameters.¶
There are also some important limitations to be aware of. For example, Prio3
provides domain separation between families of circuits, but does not provide
domain separation between instances of a circuit. Concretely, it is possible
for Aggregators to accept a report for Prio3SumVec from a Client who disagrees
with them on the value of bits
and length
. This is because there is no
binding of the circuit parameters to the computation.¶
IANA is requested to make one new registry:¶
DAF and VDAF Identifiers¶
This registry should be created under the heading "Verifiable Distributed Aggregation Functions (VDAF)", and administered under the Specification Required policy [RFC8126].¶
The "VDAF Identifiers" registry lists identifiers for Distributed Aggregation Functions (DAFs) and Verifiable Distributed Aggregation Functions (VDAFs). These identifiers are four-byte values, so the minimum possible value is 0x00000000 and the maximum possible value is 0xffffffff.¶
Template:¶
Value: The four-byte identifier for the DAF or VDAF¶
Scheme: The name of the DAF or VDAF¶
Type: Either "DAF" for a Distributed Aggregation Function or "VDAF" for a Verifiable Distributed Aggregation Function¶
Reference: Where the algorithm is defined¶
The initial contents of the registry are as follows:¶
Value | Scheme | Type | Reference |
---|---|---|---|
0x00000000
|
Reserved | n/a | RFC XXXX |
0x00000001
|
Prio3Count | VDAF | Section 7.4.1 of RFC XXXX |
0x00000002
|
Prio3Sum | VDAF | Section 7.4.2 of RFC XXXX |
0x00000003
|
Prio3SumVec | VDAF | Section 7.4.3 of RFC XXXX |
0x00000004
|
Prio3Histogram | VDAF | Section 7.4.4 of RFC XXXX |
0x00000005
|
Prio3MultihotCountVec | VDAF | Section 7.4.5 of RFC XXXX |
0x00000006
|
Poplar1 | VDAF | Section 8.4 of RFC XXXX |
0xFFFF0000 to 0xFFFFFFFF
|
Reserved for Private Use | n/a | n/a |
(RFC EDITOR: Please replace "RFC XXXX" above with the RFC number assigned to this document.)¶
VDAF identifiers are used for domain separation, as described in Section 6.2.3. Domain separation guards against failures of entropy sources, by ensuring that invocations of different VDAFs use different derived values, even if they are invoked with the same underlying random data.¶
The benefits of domain separation are undermined if different VDAFs are used with the same VDAF Identifier. The "Reserved for Private Use" code points should thus be used judiciously, because they provide no defense against such collisions. Applications SHOULD prefer the use of registered code points.¶
The impetus of this work is the success of recent deployments of predecessors of Prio3. The Mozilla Origin Telemetry project [OriginTelemetry] and the Exposure Notification Private Analytics collaboration among the Internet Security Research Group (ISRG), Google, Apple, and others [ENPA] have together aggregated data from hundreds of millions of users.¶
As the name implies, Prio3 is a descendant of the original Prio construction [CGB17]. A second iteration was deployed in the [ENPA] system, and like the VDAF described here, the ENPA system was built from techniques introduced in [BBCGGI19] that significantly improve communication cost. That system was specialized for a particular aggregation function; the goal of Prio3 is to provide the same level of generality as the original construction.¶
The security considerations in Section 9 are based largely on the security analysis of [DPRS23]. Thanks to Hannah Davis and Mike Rosulek, who lent their time to developing definitions and security proofs.¶
Thanks to Junye Chen, Henry Corrigan-Gibbs, Armando Faz-Hernández, Simon Friedberger, Tim Geoghegan, Albert Liu, Brandon Pitman, Mariana Raykova, Michael Rosenberg, Jacob Rothstein, Shan Wang, Xiao Wang, Bas Westerbaan, and Christopher Wood for useful feedback on and contributions to the spec.¶
TODO Update the reference in [TestVectors] by replacing <TODO>
with
the commit hash with the final version of the test vectors.)¶
The test vectors are available at [TestVectors]. The directory contains a set
of JSON files. Each file contains a test vector for an instance of Vdaf
(Section 5). A test vector covers sharding, preparation, aggregation, and
unsharding of each of several measurements. The test vector schema is defined
below.¶
ctx
:The application context string encoded in hexadecimal.¶
verify_key
:The verification key encoded in hexadecimal.¶
agg_param
:The aggregation parameter of type Vdaf.AggParam
.¶
prep
:A list of objects with the following schema:¶
measurement
:The measurement of type Vdaf.Measurement
.¶
nonce
:The nonce encoded in hexadecimal.¶
rand
:The sharding randomness encoded in hexadecimal.¶
public_share
:The expected public share encoded in hexadecimal.¶
input_shares
:The expected list of input shares, each incoded in hexadecimal.¶
prep_shares
:The expected list of prep shares generated by each Aggregator at each round of preparation, encoded in hexadecimal.¶
prep_messages
:The expected list of prep messages for each round of preparation, encoded in hexadecimal.¶
out_shares
:The expected list of output shares, encoded in hexadecimal.¶
agg_shares
:The expected aggregate shares encoded in hexadecimal.¶
agg_result
:The expected aggregate result of type Vdaf.AggResult
.¶
The schema also includes whatever parameters are required to instantiate the VDAF. These are listed in the subsections below.¶
shares
:The number of shares, an integer.¶