InternetDraft  VDAF  March 2023 
Barnes, et al.  Expires 14 September 2023  [Page] 
This document describes Verifiable Distributed Aggregation Functions (VDAFs), a family of multiparty 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 input that would result in an incorrect 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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This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/licenseinfo) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Revised BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Revised BSD License.¶
The ubiquity of the Internet makes it an ideal platform for measurement of largescale 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.¶
Most prior approaches to this problem fall under the rubric 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 wellknown distribution and adds it to their input before submitting to the aggregation server. The aggregation server then adds up the noisy inputs, and because it knows the distribution from whence the noise was sampled, it can estimate the true sum with reasonable precision.¶
Differentially private systems like RAPPOR are easy to deploy and provide a useful guarantee. On its own, however, DP falls short of the strongest privacy property one could hope for. Specifically, depending on the "amount" of noise a client adds to its input, it may be possible for a curious aggregator to make a reasonable guess of the input's true value. Indeed, the more noise the clients add, the less reliable will be the server's estimate of the output. Thus systems employing DP techniques alone must strike a delicate balance between privacy and utility.¶
The ideal goal for a privacypreserving measurement system is that of secure multiparty computation (MPC): No participant in the protocol should learn anything about an individual input beyond what it can deduce from the aggregate. In this document, we describe Verifiable Distributed Aggregation Functions (VDAFs) as a general class of protocols that achieve this goal.¶
VDAF schemes achieve their privacy goal by distributing the computation of the aggregate among a number of noncolluding aggregation servers. As long as a subset of the servers executes the protocol honestly, VDAFs guarantee that no input is ever accessible to any party besides the client that submitted it. At the same time, VDAFs are "verifiable" in the sense that malformed inputs that would otherwise garble the output of the computation can be detected and removed from the set of input measurements.¶
In addition to these MPCstyle security goals, VDAFs can be composed with various mechanisms for differential privacy, thereby providing the added assurance that the aggregate result itself does not leak too much information about any one measurement.¶
TODO(issue #94) Provide guidance for local and central DP and point to it here.¶
The cost of achieving these security properties is the need for multiple servers to participate in the protocol, and the need to ensure they do not collude to undermine the VDAF's privacy guarantees. Recent implementation experience has shown that practical challenges of coordinating multiple servers can be overcome. The Prio system [CGB17] (essentially a VDAF) has been deployed in systems supporting hundreds of millions of users: 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].¶
The VDAF abstraction laid out in Section 5 represents a class of multiparty protocols for privacypreserving measurement proposed in the literature. These protocols vary in their operational and security considerations, sometimes in subtle but consequential ways. This document therefore has two important goals:¶
Providing higherlevel protocols like [DAP] with a simple, uniform interface for accessing privacypreserving measurement schemes, and documenting relevant operational and security bounds for that interface:¶
This document also specifies two concrete VDAF schemes, each based on a protocol from the literature.¶
The aforementioned Prio system [CGB17] allows for the privacypreserving computation of a variety aggregate statistics. The basic idea underlying Prio is fairly simple:¶
The difficult part of this system is ensuring that the servers hold shares of a valid input, e.g., the input is an integer in a specific range. Thus Prio specifies a multiparty protocol for accomplishing this task.¶
In Section 7 we describe 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.¶
More recently, Boneh et al. [BBCGGI21] described a protocol called Poplar
for solving the t
heavyhitters problem in a privacypreserving manner. Here
each client holds a bitstring of length n
, and the goal of the aggregation
servers is to compute the set of inputs that occur at least t
times. The
core primitive used in their protocol is a specialized Distributed Point
Function (DPF) [GI14] that allows the servers to "query" their DPF shares on
any bitstring of length shorter than or equal to n
. As a result of this
query, each of the servers has an additive share of a bit indicating whether
the string is a prefix of the client's input. The protocol also specifies a
multiparty computation for verifying that at most one string among a set of
candidates is a prefix of the client's input.¶
In Section 8 we describe a VDAF called Poplar1 that implements this functionality.¶
Finally, perhaps the most complex aspect of schemes like Prio3 and Poplar1 is the process by which the clientgenerated measurements are prepared for aggregation. Because these constructions are based on secret sharing, the servers will be required to exchange some amount of information in order to verify the measurement is valid and can be aggregated. Depending on the construction, this process may require multiple round trips over the network.¶
There are applications in which this verification step may not be necessary, e.g., when the client's software is run a trusted execution environment. To support these applications, this document also defines Distributed Aggregation Functions (DAFs) as a simpler class of protocols that aim to provide the same privacy guarantee as VDAFs but fall short of being verifiable.¶
OPEN ISSUE Decide if we should give one or two example DAFs. There are natural variants of Prio3 and Poplar1 that might be worth describing.¶
The remainder of this document is organized as follows: Section 3 gives a brief overview of DAFs and VDAFs; 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 VDAFs.¶
(*) Indicates a change that breaks wire compatibility with the previous draft.¶
05:¶
prep_next()
to
prep_shares_to_prep()
.¶
04:¶
03:¶
02:¶
01:¶
prep_next()
to
prep_shares_to_prep()
. (*)¶
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. 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
bytestring constants. When comprised of printable ASCII characters, they are
written as Python 3 bytestring literals (e.g., b'some constant string'
).¶
A global constant VERSION
of type Unsigned
is defined, which algorithms are
free to use as desired. Its value SHALL be 5
.¶
This document describes algorithms for multiparty 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: Unsigned) > Bytes
returns an array of zero bytes. The length of
output
MUST be len
.¶
gen_rand(len: Unsigned) > Bytes
returns an array of random bytes. The
length of output
MUST be len
.¶
byte(int: Unsigned) > Bytes
returns the representation of int
as a byte
string. The value of int
MUST be in [0,256)
.¶
concat(parts: Vec[Bytes]) > Bytes
returns the concatenation of the input
byte strings, i.e., parts[0]  ...  parts[len(parts)1]
.¶
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: Unsigned, length: Unsigned) > Bytes
converts val
to
bigendian bytes; its value MUST be in range [0, 2^(8*length))
. Function
from_be_bytes(encoded: Bytes) > Unsigned
computes the inverse.¶
to_le_bytes(val: Unsigned, length: Unsigned) > Bytes
converts val
to
littleendian bytes; its value MUST be in range [0, 2^(8*length))
. Function
from_le_bytes(encoded: Bytes) > Unsigned
computes the inverse.¶
next_power_of_2(n: Unsigned) > Unsigned
returns the smallest integer
greater than or equal to n
that is also a power of two.¶
In a DAF or VDAFbased private measurement system, we distinguish three types of actors: Clients, Aggregators, and Collectors. The overall flow of the measurement process is as follows:¶
The Aggregators convert their input shares into "output shares".¶
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 noncollusion 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), and Collectors trust Aggregators to correctly run the protocol.¶
Within the bounds of the noncollusion 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 clientserver protocol.¶
In this document, we describe the computations performed by the actors in this system. It is up to the higherlevel 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 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

Algorithm identifier for this DAF. 
SHARES

Number of input shares into which each measurement is sharded 
RAND_SIZE

Size of the random byte string passed to sharding algorithm 
Measurement

Type of each measurement 
AggParam

Type of aggregation parameter 
OutShare

Type of each output share 
AggResult

Type of the aggregate result 
These types define some of the inputs and outputs of DAF methods at various
stages of the computation. Observe that only the measurements, output shares,
the aggregate result, and the aggregation parameter have an explicit type. All
other values  in particular, the input shares and the aggregate shares 
have type Bytes
and are treated as opaque byte strings. This is because these
values must be transmitted between parties over a network.¶
OPEN ISSUE It might be cleaner to define a type for each value, then have that type implement an encoding where necessary. This way each method parameter has a meaningful type hint. See issue#58.¶
Each DAF is identified by a unique, 32bit integer ID
. Identifiers for each
(V)DAF specified in this document are defined in Table 16.¶
In order to protect the privacy of its measurements, a DAF Client shards its
measurements into a sequence of input shares. The measurement_to_input_shares
method is used for this purpose.¶
Daf.measurement_to_input_shares(input: Measurement, rand:
Bytes[Daf.RAND_SIZE]) > (Bytes, Vec[Bytes])
is the randomized sharding
algorithm run by each Client. (The input rand
consists of the random coins
consumed by the algorithm.) It consumes the measurement and produces a "public
share", distributed to each of the Aggregators, and a corresponding sequence
of input shares, one for each Aggregator. The length of the output vector MUST
be SHARES
.¶
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(agg_id: Unsigned, agg_param: AggParam, public_share: Bytes,
input_share: Bytes) > OutShare
is the deterministic preparation algorithm.
It takes as input the public share and one of the input shares generated by a
Client, the Aggregator's unique identifier, and the aggregation parameter
selected by the Collector and returns an output share.¶
The protocol in which the DAF is used MUST ensure that the Aggregator's
identifier is equal to the integer in range [0, SHARES)
that matches the index
of input_share
in the sequence of input shares output by the Client.¶
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: Vec[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.out_shares_to_agg_share(agg_param: AggParam, out_shares: Vec[OutShare])
> agg_share: Bytes
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 "oneshot" 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 elementwise. 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.agg_shares_to_result(agg_param: AggParam,
agg_shares: Vec[Bytes], num_measurements: Unsigned) > 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.¶
QUESTION Maybe the aggregation algorithms should be randomized in order to allow the Aggregators (or the Collector) to add noise for differential privacy. (See the security considerations of [DAP].) Or is this outofscope of this document? See https://github.com/ietfwgppm/ppmspecification/issues/19.¶
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 pointtopoint 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 Clientgenerated 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:¶
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 
AggParam

Type of aggregation parameter 
Prep

State of each Aggregator during Preparation (Section 5.2) 
OutShare

Type of each output share 
AggResult

Type of the aggregate result 
Similarly to DAFs (see {[secdaf}}), any output of a VDAF method that must be transmitted from one party to another is treated as an opaque byte string. All other quantities are given a concrete type.¶
OPEN ISSUE It might be cleaner to define a type for each value, then have that type implement an encoding where necessary. See issue#58.¶
Each VDAF is identified by a unique, 32bit integer ID
. Identifiers for each
(V)DAF specified in this document are defined in Table 16. The following
method is defined for every VDAF:¶
def custom(Vdaf, usage: Unsigned) > Bytes: return format_custom(0, Vdaf.ID, usage)¶
It is used to construct a customization string for an instance of Prg
used by
the VDAF. (See Section 6.2.)¶
Sharding transforms a measurement into input shares as it does in DAFs (cf. Section 4.1); in addition, it takes a nonce as input and produces a public share:¶
Vdaf.measurement_to_input_shares(measurement: Measurement, nonce:
Bytes[Vdaf.NONCE_SIZE], rand: Bytes[Vdaf.RAND_SIZE]) > (Bytes, Vec[Bytes])
is the randomized sharding algorithm run by each Client. (Input rand
consists of the random coins consumed by the algorithm.) It consumes the
measurement and the nonce and produces a public share, distributed to each of
Aggregators, and the 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). The length of the output vector MUST be SHARES
.¶
In order to ensure privacy of the measurement, the Client MUST generate the nonce using a cryptographically secure pseudorandom number generator (CSPRNG). (See Section 9 for details.)¶
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". (Each of the outbound messages
are called "preparationmessage shares".) 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 preparation message. Its output in the last round is its output share and its output in each of the preceding rounds is a preparationmessage 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 inputs for aggregation.¶
To facilitate the preparation process, a concrete VDAF implements the following class methods:¶
Vdaf.prep_init(verify_key: Bytes[Vdaf.VERIFY_KEY_SIZE], agg_id: Unsigned,
agg_param: AggParam, nonce: Bytes[Vdaf.NONCE_SIZE], public_share: Bytes,
input_share: Bytes) > Prep
is the deterministic preparationstate
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 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.¶
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 using the VDAF MUST ensure that the Aggregator's identifier is equal
to the integer in range [0, SHARES)
that matches the index of input_share
in the sequence of input shares output by the Client.¶
Protocols MUST ensure that public share consumed by each of the Aggregators is identical. This is security critical for VDAFs such as Poplar1 that require an extractable distributed point function. (See Section 8 for details.)¶
Vdaf.prep_next(prep: Prep, inbound: Optional[Bytes]) > Union[Tuple[Prep,
Bytes], OutShare]
is the deterministic preparationstate update algorithm run
by each Aggregator. It updates the Aggregator's preparation state (prep
) 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 or, if this is the first round,
None
.¶
Vdaf.prep_shares_to_prep(agg_param: AggParam, prep_shares: Vec[Bytes]) >
Bytes
is the deterministic preparationmessage preprocessing algorithm. It
combines the preparationmessage shares generated by the Aggregators in the
previous round into the preparation message consumed by each in the next
round.¶
In effect, each Aggregator moves through a linear state machine with ROUNDS+1
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.¶
TODO Consider how to bake this "linear state machine" condition into the syntax. Given that Python 3 is used as our pseudocode, it's easier to specify the preparation state using a class.¶
The preparationstate 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.¶
Note that it is possible for a VDAF to specify ROUNDS == 0
, in which case each
Aggregator runs the preparationstate update algorithm once and immediately
recovers its output share without interacting with the other Aggregators.
However, most, if not all, constructions will require some amount of interaction
in order to ensure validity of the output shares (while also maintaining
privacy).¶
OPEN ISSUE accommodating 0round VDAFs may require syntax changes if, for example, public keys are required. On the other hand, we could consider defining this class of schemes as a different primitive. See issue#77.¶
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: Vec[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.out_shares_to_agg_share(agg_param: AggParam, out_shares: Vec[OutShare])
> agg_share: Bytes
is the deterministic aggregation algorithm. It is run by
each Aggregator over the output shares it has computed over a batch of
measurement inputs.¶
The data flow for this stage is illustrated in Figure 3. Here again, we have the aggregation algorithm in a "oneshot" 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.agg_shares_to_result(agg_param: AggParam,
agg_shares: Vec[Bytes], num_measurements: Unsigned) > 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.¶
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: Unsigned
is the prime modulus that defines the field.¶
ENCODED_SIZE: Unsigned
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: Unsigned) > output: Vec[Field]
returns a vector of
zeros. The length of output
MUST be length
.¶
Field.rand_vec(length: Unsigned) > output: Vec[Field]
returns a vector of
random field elements. The length of output
MUST be length
.¶
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() > Unsigned
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: Unsigned)
returns integer
represented as a field element.
The value of integer
MUST be less than Field.MODULUS
.¶
Finally, 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.¶
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 discrete Fourier transform, as this allows for fast polynomial interpolation. (One example is Prio3 (Section 7) when instantiated with the generic FLP of Section 7.3.3.) Specifically, a field is said to be "FFTfriendly" 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 FFTfriendly, 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
.¶
FFTfriendly fields also define the following parameter:¶
GEN_ORDER: Unsigned
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 
OPEN ISSUE We currently use bigendian for encoding field elements. However,
for implementations of GF(2^25519)
, little endian is more common. See
issue#90.¶
A pseudorandom generator (PRG) is used to expand a short, (pseudo)random seed into a long string of pseudorandom bits. A PRG suitable for this document implements the interface specified in this section.¶
PRGs are defined by a class Prg
with the following associated parameter:¶
SEED_SIZE: Unsigned
is the size (in bytes) of a seed.¶
A concrete Prg
implements the following class method:¶
Prg(seed: Bytes[Prg.SEED_SIZE], custom: Bytes, binder: Bytes)
constructs an
instance of Prg
from the given seed and customization and binder strings.
(See below for definitions of these.) The seed MUST be of length SEED_SIZE
and MUST be generated securely (i.e., it is either the output of gen_rand
or
a previous invocation of the PRG).¶
prg.next(length: Unsigned)
returns the next length
bytes of output of PRG.
If the seed was securely generated, the output can be treated as pseudorandom.¶
Each Prg
has two derived class methods. The first is used to derive a fresh
seed from an existing one. The second is used to compute a sequence of
pseudorandom field elements. For each method, the seed MUST be of length
SEED_SIZE
and MUST be generated securely (i.e., it is either the output of
gen_rand
or a previous invocation of the PRG).¶
This section describes PrgSha3, a PRG based on the Keccak permutation of SHA3 [FIPS202]. Keccak is used in the cSHAKE128 mode of operation [SP800185]. This Prg is RECOMMENDED for all use cases within VDAFs.¶
While PrgSha3 as described above can be securely used in all cases where a Prg is needed in the VDAFs described in this document, there are some cases where a more efficient instantiation based on fixedkey AES is possible. For now, this is limited to the Prg used inside the Idpf Section 8.1 implementation in Poplar1 Section 8.3. It is NOT RECOMMENDED to use this Prg anywhere else. See Security Considerations Section 9 for a more detailed discussion.¶
class PrgFixedKeyAes128(Prg): # Associated parameters SEED_SIZE = 16 def __init__(self, seed, custom, binder): self.length_consumed = 0 # Use SHA3 to derive a key from the binder and customization # strings. Note that the AES key does not need to be kept # secret from any party. However, when used with IdpfPoplar, # we require the binder to be a random nonce. # # Implementation note: This step can be cached across PRG # evaluations with many different seeds. self.fixed_key = cSHAKE128(binder, 16, b'', custom) self.seed = seed def next(self, length: Unsigned) > Bytes: offset = self.length_consumed % 16 new_length = self.length_consumed + length block_range = range( int(self.length_consumed / 16), int(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] # The multiinstance tweakable circular correlationrobust hash function of # [GKWWY20] (Section 4.2). # # Function `AES128(key, block)` is the AES128 blockcipher. def hash_block(self, block): lo, hi = block[:8], block[8:] sigma = hi + xor(hi, lo) return xor(AES128(self.fixed_key, sigma), sigma)¶
PRGs 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 "customization string". The customization string encodes the following values:¶
VERSION
);¶
The following algorithm is used in the remainder of this document in order to format the customization string:¶
def format_custom(algo_class: Unsigned, algo: Unsigned, usage: Unsigned) > Bytes: return 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:¶
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 multiparty computation to determine if their shares correspond to a valid input (as determined by the arithmetic circuit). This computation involves a "proof" of validity generated by the Client. Next, each Aggregator sums up its input 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 input share by applying a fixed map. See Section 8 for an example of a VDAF that makes meaningful use of the aggregation parameter.¶
As the name implies, Prio3 is a descendant of the original Prio construction. 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 core component of Prio3 is a "Fully Linear Proof (FLP)" system. Introduced by [BBCGGI19], the FLP encapsulates the functionality required for encoding and validating inputs. 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 twoparty 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 input's validity and distributes shares of the proof to the Aggregators. Each Aggregator then performs some a computation on its input 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 input. (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 
INPUT_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 input and proof 
Measurement

Type of the measurement 
AggResult

Type of the aggregate result 
Field

As defined in (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(input: Vec[Field], prove_rand: Vec[Field], joint_rand: Vec[Field])
> Vec[Field]
is the deterministic proofgeneration algorithm run by the
prover. Its inputs are the encoded input, 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(input: Vec[Field], proof: Vec[Field], query_rand: Vec[Field],
joint_rand: Vec[Field], num_shares: Unsigned) > Vec[Field]
is the
querygeneration algorithm run by the verifier. This is used to "query" the
input and proof. The result of the query (i.e., the output of this function)
is called the "verifier message". In addition to the input 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 input and proof shares were generated.¶
Flp.decide(verifier: Vec[Field]) > Bool
is the deterministic decision
algorithm run by the verifier. It takes as input the verifier message and
outputs a boolean indicating if the input 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 input and proof, the querygeneration algorithm outputs a share of the verifier message. Furthermore, the "strong zeroknowledge" property of the FLP system ensures that the verifier message reveals nothing about the input's validity. Therefore, to decide if an input is valid, the Aggregators will run the querygeneration algorithm locally, exchange verifier shares, combine them to recover the verifier message, and run the decision algorithm.¶
The querygeneration algorithm includes a parameter num_shares
that specifies
the number of shares of the input and proof 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 input, 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 input
is a valid input, then
run_flp(Flp, input, 1)
always returns True
. On the other hand, if input
is
invalid, then as long as joint_rand
and query_rand
are generated uniform
randomly, the output is False
with overwhelming probability.¶
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 1.5round, publiccoin, 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) > Vec[Field]
encodes a raw measurement
as a vector of field elements. The return value MUST be of length INPUT_LEN
.¶
For some FLPs, the encoded input 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 [0, 2^k)
is encoded as a vector of k
field elements (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 input to the length of the aggregated output:¶
Flp.truncate(input: Vec[Field]) > Vec[Field]
maps an encoded input to an
aggregatable output. The length of the input MUST be INPUT_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 postprocessing.¶
Flp.decode(output: Vec[Field], num_measurements: Unsigned) > AggResult
maps a sum of aggregate shares to an aggregate result. The length of the
input MUST be OUTPUT_LEN
. num_measurements
is the number of measurements
that contributed to the aggregated output.¶
We remark that, taken together, these three functionalities correspond roughly to the notion of "Affineaggregatable encodings (AFEs)" from [CGB17].¶
This section specifies Prio3
, an implementation of the Vdaf
interface
(Section 5). It has two generic parameters: an Flp
(Section 7.1) and a Prg
(Section 6.2). 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

Prg.SEED_SIZE

NONCE_SIZE

16

ROUNDS

1

SHARES

in [2, 256)

Measurement

Flp.Measurement

AggParam

None

Prep

Tuple[Vec[Flp.Field], Optional[Bytes], Bytes]

OutShare

Vec[Flp.Field]

AggResult

Flp.AggResult

Variable  Value 

DST_MEASUREMENT_SHARE: Unsigned

1 
DST_PROOF_SHARE: Unsigned

2 
DST_JOINT_RANDOMNESS: Unsigned

3 
DST_PROVE_RANDOMNESS: Unsigned

4 
DST_QUERY_RANDOMNESS: Unsigned

5 
DST_JOINT_RAND_SEED: Unsigned

6 
DST_JOINT_RAND_PART: Unsigned

7 
This section describes the process of recovering output shares from the input shares. The highlevel idea is that each Aggregator first queries its input and proof share locally, then exchanges its verifier share with the other Aggregators. The verifier shares are then combined into the verifier message, which is used to decide whether to accept.¶
In addition, the Aggregators must ensure that they have all used the same joint randomness for the querygeneration algorithm. The joint randomness is generated by a PRG seed. Each Aggregator derives a "part" of this seed from its input share and the "blind" generated by the client. The seed is derived by hashing together the parts, so before running the querygeneration algorithm, it must first gather the parts derived by the other Aggregators.¶
In order to avoid extra round of communication, the Client sends each Aggregator a "hint" consisting of the other Aggregators' parts of the joint randomness seed. 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 verifier shares.¶
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 elementwise.¶
To unshard a set of aggregate shares, the Collector first adds up the vectors elementwise. It then converts each element of the vector into an integer.¶
def joint_rand(Prio3, k_joint_rand_parts): return Prio3.Prg.derive_seed( zeros(Prio3.Prg.SEED_SIZE), Prio3.custom(DST_JOINT_RAND_SEED), concat(k_joint_rand_parts), )¶
def encode_leader_share(Prio3, meas_share, proof_share, k_blind): encoded = Bytes() encoded += Prio3.Flp.Field.encode_vec(meas_share) encoded += Prio3.Flp.Field.encode_vec(proof_share) if Prio3.Flp.JOINT_RAND_LEN > 0: encoded += k_blind return encoded def decode_leader_share(Prio3, encoded): l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.INPUT_LEN encoded_meas_share, encoded = encoded[:l], encoded[l:] meas_share = Prio3.Flp.Field.decode_vec(encoded_meas_share) l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.PROOF_LEN encoded_proof_share, encoded = encoded[:l], encoded[l:] proof_share = Prio3.Flp.Field.decode_vec(encoded_proof_share) l = Prio3.Prg.SEED_SIZE if Prio3.Flp.JOINT_RAND_LEN == 0: if len(encoded) != 0: raise ERR_DECODE return (meas_share, proof_share, None) k_blind, encoded = encoded[:l], encoded[l:] if len(encoded) != 0: raise ERR_DECODE return (meas_share, proof_share, k_blind) def encode_helper_share(Prio3, k_meas_share, k_proof_share, k_blind): encoded = Bytes() encoded += k_meas_share encoded += k_proof_share if Prio3.Flp.JOINT_RAND_LEN > 0: encoded += k_blind return encoded def decode_helper_share(Prio3, agg_id, encoded): c_meas_share = Prio3.custom(DST_MEASUREMENT_SHARE) c_proof_share = Prio3.custom(DST_PROOF_SHARE) l = Prio3.Prg.SEED_SIZE k_meas_share, encoded = encoded[:l], encoded[l:] meas_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field, k_meas_share, c_meas_share, byte(agg_id), Prio3.Flp.INPUT_LEN) k_proof_share, encoded = encoded[:l], encoded[l:] proof_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field, k_proof_share, c_proof_share, byte(agg_id), Prio3.Flp.PROOF_LEN) if Prio3.Flp.JOINT_RAND_LEN == 0: if len(encoded) != 0: raise ERR_DECODE return (meas_share, proof_share, None) k_blind, encoded = encoded[:l], encoded[l:] if len(encoded) != 0: raise ERR_DECODE return (meas_share, proof_share, k_blind) def encode_public_share(Prio3, k_joint_rand_parts): encoded = Bytes() if Prio3.Flp.JOINT_RAND_LEN > 0: encoded += concat(k_joint_rand_parts) return encoded def decode_public_share(Prio3, encoded): l = Prio3.Prg.SEED_SIZE if Prio3.Flp.JOINT_RAND_LEN == 0: if len(encoded) != 0: raise ERR_DECODE return None k_joint_rand_parts = [] for i in range(Prio3.SHARES): k_joint_rand_part, encoded = encoded[:l], encoded[l:] k_joint_rand_parts.append(k_joint_rand_part) if len(encoded) != 0: raise ERR_DECODE return k_joint_rand_parts def encode_prep_share(Prio3, verifier, k_joint_rand): encoded = Bytes() encoded += Prio3.Flp.Field.encode_vec(verifier) if Prio3.Flp.JOINT_RAND_LEN > 0: encoded += k_joint_rand return encoded def decode_prep_share(Prio3, encoded): l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.VERIFIER_LEN encoded_verifier, encoded = encoded[:l], encoded[l:] verifier = Prio3.Flp.Field.decode_vec(encoded_verifier) if Prio3.Flp.JOINT_RAND_LEN == 0: if len(encoded) != 0: raise ERR_DECODE return (verifier, None) l = Prio3.Prg.SEED_SIZE k_joint_rand, encoded = encoded[:l], encoded[l:] if len(encoded) != 0: raise ERR_DECODE return (verifier, k_joint_rand) def encode_prep_msg(Prio3, k_joint_rand_check): encoded = Bytes() if Prio3.Flp.JOINT_RAND_LEN > 0: encoded += k_joint_rand_check return encoded def decode_prep_msg(Prio3, encoded): if Prio3.Flp.JOINT_RAND_LEN == 0: if len(encoded) != 0: raise ERR_DECODE return None l = Prio3.Prg.SEED_SIZE k_joint_rand_check, encoded = encoded[:l], encoded[l:] if len(encoded) != 0: raise ERR_DECODE return k_joint_rand_check¶
This section describes an FLP based on the construction from in [BBCGGI19], Section 4.2. We begin in Section 7.3.1 with an overview of their proof system and the extensions to their proof system made here. The construction is specified in Section 7.3.3.¶
OPEN ISSUE We're not yet sure if specifying this generalpurpose FLP is desirable. It might be preferable to specify specialized FLPs for each data type that we want to standardize, for two reasons. First, clear and concise specifications are likely easier to write for specialized FLPs rather than the general one. Second, we may end up tailoring each FLP to the measurement type in a way that improves performance, but breaks compatibility with the generalpurpose FLP.¶
In any case, we can't make this decision until we know which data types to standardize, so for now, we'll stick with the generalpurpose construction. The reference implementation can be found at https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc.¶
OPEN ISSUE Chris Wood points out that the this section reads more like a paper than a standard. Eventually we'll want to work this into something that is readily consumable by the CFRG.¶
In the proof system of [BBCGGI19], validity is defined via an arithmetic circuit evaluated over the input: If the circuit output is zero, then the input is deemed valid; otherwise, if the circuit output is nonzero, then the input is deemed invalid. Thus the goal of the proof system is merely to allow the verifier to evaluate the validity circuit over the input. For our application (Section 7), this computation is distributed among multiple Aggregators, each of which has only a share of the input.¶
Suppose for a moment that the validity circuit C
is affine, meaning its only
operations are addition and multiplicationbyconstant. In particular, suppose
the circuit does not contain a multiplication gate whose operands are both
nonconstant. Then to decide if an input x
is valid, each Aggregator could
evaluate C
on its share of x
locally, broadcast the output share to its
peers, then combine the output shares locally to recover C(x)
. This is true
because for any SHARES
way secret sharing of x
it holds that¶
C(x_shares[0] + ... + x_shares[SHARES1]) = C(x_shares[0]) + ... + C(x_shares[SHARES1])¶
(Note that, for this equality to hold, it may be necessary to scale any
constants in the circuit by SHARES
.) However this is not the case if C
is
notaffine (i.e., it contains at least one multiplication gate whose operands
are nonconstant). In the proof system of [BBCGGI19], the proof is designed to
allow the (distributed) verifier to compute the nonaffine operations using only
linear operations on (its share of) the input and proof.¶
To make this work, the proof system is restricted to validity circuits that
exhibit a special structure. Specifically, an arithmetic circuit with "Ggates"
(see [BBCGGI19], Definition 5.2) is composed of affine gates and any number of
instances of a distinguished gate G
, which may be nonaffine. We will refer to
this class of circuits as 'gadget circuits' and to G
as the "gadget".¶
As an illustrative example, consider a validity circuit C
that recognizes the
set L = set([0], [1])
. That is, C
takes as input a length1 vector x
and
returns 0 if x[0]
is in [0,2)
and outputs something else otherwise. This
circuit can be expressed as the following degree2 polynomial:¶
C(x) = (x[0]  1) * x[0] = x[0]^2  x[0]¶
This polynomial recognizes L
because x[0]^2 = x[0]
is only true if x[0] ==
0
or x[0] == 1
. Notice that the polynomial involves a nonaffine operation,
x[0]^2
. In order to apply [BBCGGI19], Theorem 4.3, the circuit needs to be
rewritten in terms of a gadget that subsumes this nonaffine operation. For
example, the gadget might be multiplication:¶
Mul(left, right) = left * right¶
The validity circuit can then be rewritten in terms of Mul
like so:¶
C(x[0]) = Mul(x[0], x[0])  x[0]¶
The proof system of [BBCGGI19] allows the verifier to evaluate each instance
of the gadget (i.e., Mul(x[0], x[0])
in our example) using a linear function
of the input and proof. The proof is constructed roughly as follows. Let C
be
the validity circuit and suppose the gadget is arityL
(i.e., it has L
input
wires.). Let wire[j1,k1]
denote the value of the j
th wire of the k
th
call to the gadget during the evaluation of C(x)
. Suppose there are M
such
calls and fix distinct field elements alpha[0], ..., alpha[M1]
. (We will
require these points to have a special property, as we'll discuss in
Section 7.3.1.1; but for the moment it is only important
that they are distinct.)¶
The prover constructs from wire
and alpha
a polynomial that, when evaluated
at alpha[k1]
, produces the output of the k
th call to the gadget. Let us
call this the "gadget polynomial". Polynomial evaluation is linear, which means
that, in the distributed setting, the Client can disseminate additive shares of
the gadget polynomial that the Aggregators then use to compute additive shares
of each gadget output, allowing each Aggregator to compute its share of C(x)
locally.¶
There is one more wrinkle, however: It is still possible for a malicious prover
to produce a gadget polynomial that would result in C(x)
being computed
incorrectly, potentially resulting in an invalid input being accepted. To
prevent this, the verifier performs a probabilistic test to check that the
gadget polynomial is wellformed. This test, and the procedure for constructing
the gadget polynomial, are described in detail in Section 7.3.3.¶
The FLP described in the next section extends the proof system [BBCGGI19], Section 4.2 in three ways.¶
First, the validity circuit in our construction includes an additional, random
input (this is the "joint randomness" derived from the input 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 lengthN
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(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N1])¶
(Note that this is a special case of [BBCGGI19], Theorem 5.2.) Here inp
is
the lengthN
input and r
is a random field element. The gadget circuit
Range2
is the "rangecheck" polynomial described above, i.e., Range2(x) = x^2 
x
. The idea is that, if inp
is valid (i.e., each inp[j]
is in [0,2)
),
then the circuit will evaluate to 0 regardless of the value of r
; but if
inp[j]
is not in [0,2)
for some j
, the output will be nonzero 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.) For example, the following circuit is allowed, where
Mul
and Range2
are the gadgets defined above (the input has length N+1
):¶
C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N1]) + \ 2^0 * inp[0] + ... + 2^(N1) * inp[N1]  \ Mul(inp[N], inp[N])¶
Finally, [BBCGGI19], Theorem 4.3 makes no restrictions on the choice of the
fixed points alpha[0], ..., alpha[M1]
, other than to require that the points
are distinct. In this document, the fixed points are chosen so that the gadget
polynomial can be constructed efficiently using the CooleyTukey FFT ("Fast
Fourier Transform") algorithm. Note that this requires the field to be
"FFTfriendly" as defined in Section 6.1.2.¶
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 
INPUT_LEN

Length of the input 
OUTPUT_LEN

Length of the aggregatable output 
JOINT_RAND_LEN

Length of the random input 
Measurement

The type of measurement 
AggResult

Type of the aggregate result 
Field

An FFTfriendly finite field as defined in Section 6.1.2 
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).¶
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) > Vec[Field]
returns a vector of
length INPUT_LEN
representing a measurement.¶
Valid.truncate(input: Vec[Field]) > Vec[Field]
returns a vector of length
OUTPUT_LEN
representing an aggregatable output.¶
Valid.decode(output: Vec[Field], num_measurements: Unsigned) > AggResult
returns an aggregate result.¶
Finally, the following class methods are derived for each concrete Valid
:¶
This section specifies FlpGeneric
, 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.¶
NOTE A reference implementation can be found in https://github.com/cfrg/draftirtfcfrgvdaf/blob/main/poc/flp_generic.sage.¶
The FLP parameters for FlpGeneric
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

INPUT_LEN

Valid.INPUT_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

Field

Valid.Field

On input inp
, prove_rand
, and joint_rand
, the proof is computed as
follows:¶
i
in [H]
create an empty table wire_i
.¶
prove_rand
into subvectors 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.¶
Valid
on input of inp
and joint_rand
, recording the inputs of
each gadget in the corresponding table. Specifically, for every i
in [H]
,
set wire_i[j1,k1]
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[j1]
, the j
th wire polynomial for the
i
th gadget, as follows:¶
w = [seed_i[j1], wire_i[j1,0], ..., wire_i[j1,M_i1]]
.¶
padded_w = w + Field.zeros(P_i  len(w))
.¶
NOTE We pad w
to the nearest power of 2 so that we can use FFT for
interpolating the wire polynomials. Perhaps there is some clever math for
picking wire_inp
in a way that avoids having to pad.¶
poly_wire_i[j1]
be the lowest degree polynomial for which
poly_wire_i[j1](alpha_i^k) == padded_w[k]
for all k
in [P_i]
.¶
Compute the "gadget polynomials". That is, for every i
in [H]
:¶
poly_gadget_i = G_i(poly_wire_i[0], ..., poly_wire_i[L_i1])
. 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 inp
, proof
, query_rand
, and joint_rand
, the verifier message
is generated as follows:¶
i
in [H]
create an empty table wire_i
.¶
proof
into the subvectors seed_1
, coeff_1
, ..., seed_H
,
coeff_H
defined in Section 7.3.3.1.¶
Valid
on input of inp
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 v
denote the output of the circuit
evaluation.¶
Compute the tests for wellformedness 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]
.¶
This section specifies instantiations of Prio3 for various measurement types.
Each uses FlpGeneric
as the FLP (Section 7.3) and is determined by a
validity circuit (Section 7.3.2) and a PRG (Section 6.2). Test vectors for
each can be found in Appendix "Test Vectors".¶
NOTE Reference implementations of each of these VDAFs can be found in https://github.com/cfrg/draftirtfcfrgvdaf/blob/main/poc/vdaf_prio3.sage.¶
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.¶
This instance uses PrgSha3 (Section 6.2.1) as its PRG. Its validity
circuit, denoted Count
, uses Field64
(Table 3) as its finite field. Its
gadget, denoted Mul
, is the degree2, arity2 gadget defined as¶
def Mul(x, y): return x * y¶
The validity circuit is defined as¶
def Count(inp: Vec[Field64]): return Mul(inp[0], inp[0])  inp[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]

INPUT_LEN

1

OUTPUT_LEN

1

JOINT_RAND_LEN

0

Measurement

Unsigned , in range [0,2)

AggResult

Unsigned

Field

Field64 (Table 3) 
The next instance of Prio3 supports summing of integers in a predetermined
range. Each measurement is an integer in range [0, 2^bits)
, where bits
is an
associated parameter.¶
This instance of Prio3 uses PrgSha3 (Section 6.2.1) as its PRG. Its validity
circuit, denoted Sum
, uses Field128
(Table 3) as its finite field. The
measurement is encoded as a lengthbits
vector of field elements, where the
l
th element of the vector represents the l
th bit of the summand:¶
def encode(Sum, measurement: Integer): if 0 > measurement or measurement >= 2^Sum.INPUT_LEN: raise ERR_INPUT encoded = [] for l in range(Sum.INPUT_LEN): encoded.append(Sum.Field((measurement >> l) & 1)) return encoded def truncate(Sum, inp): decoded = Sum.Field(0) for (l, b) in enumerate(inp): w = Sum.Field(1 << l) decoded += w * b return [decoded] def decode(Sum, output, _num_measurements): return output[0].as_unsigned()¶
The validity circuit checks that the input consists of ones and zeros. Its
gadget, denoted Range2
, is the degree2, arity1 gadget defined as¶
def Range2(x): return x^2  x¶
The validity circuit is defined as¶
def Sum(inp: Vec[Field128], joint_rand: Vec[Field128]): out = Field128(0) r = joint_rand[0] for x in inp: out += r * Range2(x) r *= joint_rand[0] return out¶
Parameter  Value 

GADGETS

[Range2]

GADGET_CALLS

[bits]

INPUT_LEN

bits

OUTPUT_LEN

1

JOINT_RAND_LEN

1

Measurement

Unsigned , in range [0, 2^bits)

AggResult

Unsigned

Field

Field128 (Table 3) 
This instance of Prio3 allows for estimating the distribution of the measurements by computing a simple histogram. Each measurement is an arbitrary integer and the aggregate result counts the number of measurements that fall in a set of fixed buckets.¶
This instance of Prio3 uses PrgSha3 (Section 6.2.1) as its PRG. Its validity
circuit, denoted Histogram
, uses Field128
(Table 3) as its finite
field. The measurement is encoded as a onehot vector representing the bucket
into which the measurement falls (let bucket
denote a sequence of
monotonically increasing integers):¶
def encode(Histogram, measurement: Integer): boundaries = buckets + [Infinity] encoded = [Field128(0) for _ in range(len(boundaries))] for i in range(len(boundaries)): if measurement <= boundaries[i]: encoded[i] = Field128(1) return encoded def truncate(Histogram, inp: Vec[Field128]): return inp def decode(Histogram, output: Vec[Field128], _num_measurements): return [bucket_count.as_unsigned() for bucket_count in output]¶
The validity circuit uses Range2
(see Section 7.4.2) as its single gadget. It
checks for onehotness in two steps, as follows:¶
def Histogram(inp: Vec[Field128], joint_rand: Vec[Field128], num_shares: Unsigned): # Check that each bucket is one or zero. range_check = Field128(0) r = joint_rand[0] for x in inp: range_check += r * Range2(x) r *= joint_rand[0] # Check that the buckets sum to 1. sum_check = Field128(1) * Field128(num_shares).inv() for b in inp: sum_check += b out = joint_rand[1] * range_check + \ joint_rand[1]^2 * sum_check return out¶
Note that this circuit depends on the number of shares into which the input is sharded. This is provided to the FLP by Prio3.¶
Parameter  Value 

GADGETS

[Range2]

GADGET_CALLS

[buckets + 1]

INPUT_LEN

buckets + 1

OUTPUT_LEN

buckets + 1

JOINT_RAND_LEN

2

Measurement

Integer

AggResult

Vec[Unsigned]

Field

Field128 (Table 3) 
This section specifies Poplar1, a VDAF for the following task. Each Client holds
a string of length BITS
and the Aggregators hold a set 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 inputs 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.¶
0
and
1
.¶
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
heavyhitters.
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 nonzero 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 subtree 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 for details.)¶
Poplar1 composes an IDPF with the "secure sketching" protocol of [BBCGGI21]. This protocol ensures that evaluating a set of input shares on a unique set of candidate prefixes results in shares of a "onehot" vector, i.e., a vector that is zero everywhere except for 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 constant
defined by the IDPF. Indexes into the IDPF tree are encoded as integers in range
[0, 2^BITS)
. The Client specifies an index alpha
and a vector of
values beta
, one for each "level" L
in range [0, BITS)
. The key generation
algorithm generates one IDPF "key" for each Aggregator. When evaluated at level
L
and index 0 <= prefix < 2^L
, each IDPF key returns an additive share of
beta[L]
if prefix
is the L
bit prefix of alpha
and shares of zero
otherwise.¶
An index x
is defined to be a prefix of another index y
as follows. Let
LSB(x, N)
denote the least significant N
bits of positive integer x
. By
definition, a positive integer 0 <= x < 2^L
is said to be the lengthL
prefix of positive integer 0 <= y < 2^BITS
if LSB(x, L)
is equal to the most
significant L
bits of LSB(y, BITS)
, For example, 6 (110 in binary) is the
length3 prefix of 25 (11001), but 7 (111) is not.¶
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
Idpf.FieldInner
), and one for leaf nodes (Idpf.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 12. In
the remainder we write Idpf.Vec
as shorthand for the type
Union[Vec[Vec[Idpf.FieldInner]], Vec[Vec[Idpf.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: Unsigned, beta_inner: Vec[Vec[Idpf.FieldInner]], beta_leaf:
Vec[Idpf.FieldLeaf], binder: Bytes, rand: Bytes[Idpf.RAND_SIZE]) > (Bytes,
Vec[Bytes])
is the randomized IDPFkey generation algorithm. (Input rand
consists of the random coins it consumes.) Its inputs are the index alpha
the values beta
, and a binder string. The value of alpha
MUST be in range
[0, 2^BITS)
. The output is a public part that is sent to all Aggregators
and a vector of private IDPF keys, one for each aggregator.¶
Idpf.eval(agg_id: Unsigned, public_share: Bytes, key: Bytes, level:
Unsigned, prefixes: Vec[Unsigned], binder: Bytes) > Idpf.Vec
is the
deterministic, stateless IDPFkey 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,
and a binder string. It returns the share of the value corresponding to each
candidate prefix.¶
The output type depends on the value of level
: If level < Idpf.BITS1
, the
output is the value for an inner node, which has type
Vec[Vec[Idpf.FieldInner]]
; otherwise, if level == Idpf.BITS1
, then the
output is the value for a leaf node, which has type
Vec[Vec[Idpf.FieldLeaf]]
.¶
The value of level
MUST be in range [0, BITS)
. The indexes in prefixes
MUST all be distinct and in range [0, 2^level)
.¶
Applications MUST ensure that the Aggregator's identifier is equal to the
integer in range [0, SHARES)
that matches the index of key
in the sequence
of IDPF keys output by the Client.¶
In addition, the following method is derived for each concrete Idpf
:¶
def current_field(Idpf, level): return Idpf.FieldInner if level < Idpf.BITS1 \ else Idpf.FieldLeaf¶
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 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 IDPFkey 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 IDPFkey generator 
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 
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
Idpf.SHARES == 2
and Idpf.VALUE_LEN == 2
and an implementation of Prg
(Section 6.2). The associated constants and types required by the Vdaf
interface
are defined in Table 13. The methods required for sharding,
preparation, aggregation, and unsharding are described in the remaining
subsections. These methods make use of constants defined in Table 14.¶
Parameter  Value 

VERIFY_KEY_SIZE

Prg.SEED_SIZE

NONCE_SIZE

16

ROUNDS

2

SHARES

2

Measurement

Unsigned

AggParam

Tuple[Unsigned, Vec[Unsigned]]

Prep

Tuple[Bytes, Unsigned, Idpf.Vec]

OutShare

Idpf.Vec

AggResult

Vec[Unsigned]

Variable  Value 

DST_SHARD_RAND: Unsigned  1 
DST_CORR_INNER: Unsigned  2 
DST_CORR_LEAF: Unsigned  3 
DST_VERIFY_RAND: Unsigned  4 
The client's input is an IDPF index, denoted alpha
. The programmed IDPF values
are pairs of field elements (1, k)
where each k
is chosen at random. This
random value is used as part of the secure sketching protocol of [BBCGGI21],
Appendix C.4. After evaluating their IDPF key shares on a given sequence of
candidate prefixes, the sketching protocol is used by the Aggregators to verify
that they hold shares of a onehot vector. In addition, for each level of the
tree, the prover generates random elements a
, b
, and c
and computes¶
A = 2*a + k B = a^2 + b  k*a + c¶
and sends additive shares of a
, b
, c
, A
and B
to the Aggregators.
Putting everything together, the sharding algorithm is defined as
follows. Function encode_input_shares
is defined in Section 8.2.6.¶
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 onehot 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 onehot 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.¶
The algorithms below make use of the auxiliary function decode_input_share()
defined in Section 8.2.6.¶
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.¶
Aggregation involves simply adding up the output shares.¶
Finally, the Collector unshards the aggregate result by adding up the aggregate shares.¶
This section defines methods for serializing input shares, as required by the
Vdaf
interface. Optional serialization of the aggregation parameter is also
specified below.¶
Implementation note: The aggregation parameter includes the level of the IDPF tree and the sequence of indices to evaluate. For implementations that perform perreport 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.¶
def encode_input_shares(Poplar1, keys, corr_seed, corr_inner, corr_leaf): input_shares = [] for (key, seed, inner, leaf) in zip(keys, corr_seed, corr_inner, corr_leaf): encoded = Bytes() encoded += key encoded += seed encoded += Poplar1.Idpf.FieldInner.encode_vec(inner) encoded += Poplar1.Idpf.FieldLeaf.encode_vec(leaf) input_shares.append(encoded) return input_shares def decode_input_share(Poplar1, encoded): l = Poplar1.Idpf.KEY_SIZE key, encoded = encoded[:l], encoded[l:] l = Poplar1.Prg.SEED_SIZE corr_seed, encoded = encoded[:l], encoded[l:] l = Poplar1.Idpf.FieldInner.ENCODED_SIZE \ * 2 * (Poplar1.Idpf.BITS  1) encoded_corr_inner, encoded = encoded[:l], encoded[l:] corr_inner = Poplar1.Idpf.FieldInner.decode_vec( encoded_corr_inner) l = Poplar1.Idpf.FieldLeaf.ENCODED_SIZE * 2 encoded_corr_leaf, encoded = encoded[:l], encoded[l:] corr_leaf = Poplar1.Idpf.FieldLeaf.decode_vec( encoded_corr_leaf) if len(encoded) != 0: raise ERR_INPUT return (key, corr_seed, corr_inner, corr_leaf) def encode_agg_param(Poplar1, level, prefixes): if level > 2^16  1: raise ERR_INPUT # level too deep if len(prefixes) > 2^32  1: raise ERR_INPUT # too many prefixes encoded = Bytes() encoded += to_be_bytes(level, 2) encoded += to_be_bytes(len(prefixes), 4) packed = 0 for (i, prefix) in enumerate(prefixes): packed = prefix << ((level+1) * i) l = floor(((level+1) * len(prefixes) + 7) / 8) encoded += to_be_bytes(packed, l) return encoded def decode_agg_param(Poplar1, encoded): encoded_level, encoded = encoded[:2], encoded[2:] level = from_be_bytes(encoded_level) encoded_prefix_count, encoded = encoded[:4], encoded[4:] prefix_count = from_be_bytes(encoded_prefix_count) l = floor(((level+1) * prefix_count + 7) / 8) encoded_packed, encoded = encoded[:l], encoded[l:] packed = from_be_bytes(encoded_packed) prefixes = [] m = 2^(level+1)  1 for i in range(prefix_count): prefixes.append(packed >> ((level+1) * i) & m) if len(encoded) != 0: raise ERR_INPUT return (level, prefixes)¶
TODO(issue#32) Consider replacing the generic Prg
object here with some
fixedkey mode for AES (something along the lines of ia.cr/2019/074). This
would allow us to take advantage of hardware acceleration, which would
significantly improve performance. We use SHA3 primarily to instantiate
random oracles, but the random oracle model may not be required for IDPF. More
investigation is needed.¶
In this section we specify a concrete IDPF, called IdpfPoplar, suitable for instantiating Poplar1. The scheme gets its name from the name of the protocol of [BBCGGI21].¶
TODO We should consider giving IdpfPoplar
a more distinctive name.¶
The constant and type definitions required by the Idpf
interface are given in
Table 15.¶
IdpfPoplar requires a PRG for deriving the output shares, as well as a variety of other artifacts used internally. For performance reasons, we instantiate this object using PrgFixedKeyAes128 (Section 6.2.2). See Section 9.5 for justification of this choice.¶
Parameter  Value 

SHARES 
2

BITS  any positive integer 
VALUE_LEN  any positive integer 
KEY_SIZE 
Prg.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 IDPFkey generation algorithm makes use of auxiliary
functions extend()
, convert()
, and encode_public_share()
defined in
Section 8.3.3. In the following, we let Field2
denote the
field GF(2)
.¶
TODO Describe in prose how IDPFkey evaluation algorithm works.¶
The description of the IDPFevaluation algorithm makes use of auxiliary
functions extend()
, convert()
, and decode_public_share()
defined in
Section 8.3.3.¶
Here, pack_bits()
takes a list of bits, packs each group of eight bits into a
byte, in LSB to MSB order, padding the most significant bits of the last byte
with zeros as necessary, and returns the byte array. unpack_bits()
performs
the reverse operation: it takes in a byte array and a number of bits, and
returns a list of bits, extracting eight bits from each byte in turn, in LSB to
MSB order, and stopping after the requested number of bits. If the byte array
has an incorrect length, or if unused bits in the last bytes are not zero, it
throws an error.¶
By default, Poplar1 is instantiated with IdpfPoplar (VALUE_LEN == 2
) and
PrgSha3 (Section 6.2.1). This VDAF is suitable for any positive value of BITS
.
Test vectors can be found in Appendix "Test Vectors".¶
VDAFs have two essential security goals:¶
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:¶
Establishing secure channels:¶
In such an environment, a VDAF provides the highlevel 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 mitigate 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.¶
VDAFs do not inherently provide differential privacy [Dwo06]. The VDAF approach to private measurement can be viewed as complementary to differential privacy, relying on noncollusion instead of statistical noise to protect the privacy of the inputs. It is possible that a future VDAF could incorporate differential privacy features, e.g., by injecting noise before the sharding stage and removing it after unsharding.¶
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:¶
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 oneway.) 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 nonrepeating. For example, to achieve differential privacy it is necessary to avoid "over exposing" a measurement 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 reuse 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 nonheavyhitter values to share a common prefix, which would be leaked by a prefix tree with a sufficiently small threshold.¶
The only known, generalpurpose approach to mitigating this leakage is via differential privacy.¶
The objects we describe in Section 6.2 share a common interface, which we have called Prg. However, these are not necessarily all modeled as cryptographic Pseudorandom Generators in the security analyses of our protocols. Instead, most of them are modeled as random oracles. For these use cases, we want to be conservative in our assumptions, and hence prescribe PrgSha3 as the only RECOMMENDED Prg instantiation.¶
The one exception is the PRG used in the Idpf implementation IdpfPoplar Section 8.3. Here, a random oracle is not needed to prove security, and hence a construction based on fixedkey AES Section 6.2.2 can be used. However, as PrgFixedKeyAes128 has been shown to be differentiable from a random oracle [GKWWY20], it is NOT RECOMMENDED to use it anywhere else.¶
OPEN ISSUE: We may want to drop the common interface for PRGs and random oracles. See issue #159.¶
A codepoint for each (V)DAF in this document is defined in the table below. Note
that 0xFFFF0000
through 0xFFFFFFFF
are reserved for private use.¶
Value  Scheme  Type  Reference 

0x00000000

Prio3Count  VDAF  Section 7.4.1 
0x00000001

Prio3Sum  VDAF  Section 7.4.2 
0x00000002

Prio3Histogram  VDAF  Section 7.4.3 
0x00000003 to 0x00000FFF

reserved for Prio3  VDAF  n/a 
0x00001000

Poplar1  VDAF  Section 8.4 
0xFFFF0000 to 0xFFFFFFFF

reserved  n/a  n/a 
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 Henry CorriganGibbs, Armando FazHernández, Simon Friedberger, Tim Geoghegan, Mariana Raykova, Jacob Rothstein, Xiao Wang, and Christopher Wood for useful feedback on and contributions to the spec.¶
NOTE Machinereadable test vectors can be found at https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc/test_vec.¶
Test vectors cover the generation of input shares and the conversion of input
shares into output shares. Vectors specify the verification key, measurements,
aggregation parameter, and any parameters needed to construct the VDAF. (For
example, for Prio3Sum
, the user specifies the number of bits for representing
each summand.)¶
Byte strings are encoded in hexadecimal. To make the tests deterministic, the
random inputs of randomized algorithms were fixed to the byte sequence starting
with 0
, incrementing by 1
, and wrapping at 256
:¶
0, 1, 2, ..., 255, 0, 1, 2, ...¶
verify_key: "000102030405060708090a0b0c0d0e0f" upload_0: measurement: 1 nonce: "000102030405060708090a0b0c0d0e0f" public_share: > input_share_0: > e7a225b76420dd6dd0682380363bd782c8ca9ace6e7abc559dd873bafb503e3cc8b7 9f3b2b8b0a14676172e46be2ce2f input_share_1: > 000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f round_0: prep_share_0: > 56df20acd68725eab85e44c706629df555d635a5b899df767fe28effd0c45f60 prep_share_1: > ab20df532878da1518b04178e08a39d0d40fdbfe1876e479dc5e94b43afc68d5 prep_message: > out_share_0:  e7a225b76420dd6d out_share_1:  1b5dda489adf2292 agg_share_0: > e7a225b76420dd6d agg_share_1: > 1b5dda489adf2292 agg_result: 1¶
bits: 8 verify_key: "000102030405060708090a0b0c0d0e0f" upload_0: measurement: 100 nonce: "000102030405060708090a0b0c0d0e0f" public_share: > da6f5f000d95651e628464f5d086fde6a544464ac776c3812e2299cce1d29095 input_share_0: > 2d97ed6ebc5f6748db3017211c7a31ceeba9500b8317a7f61578b85c08a7f7d95c63 7bb14b95c062c74bd5b08625bdafbfd91fd01d22291b22b76e54a9be1f7e7015a396 ae0d6d034b3d22795b3d98db2a9aa51b4ea56c247496d380367deb786a9b34d5cb88 f0f43e5cb4ab71910c8fa20b0b518f7dd9ce11ec9bf4ce460c81d3e9db05c34e0f05 14d53f5eadea231dbc1fc575ad52e01622671eaf49a66533a6511710eeea024df0da 7e7d21451ff5fd9d38e36eee8b78bf7c0767b91e747c6b80d6ca6b726286edebeae4 ff377040d87f84288f4d0e5960e27b79387caef816c27481be166b71d38d1695e5e5 061c26f0f667f8fff3133c2ae3ac1853ac542a9af0b5e422764609abd882dedf6aa8 c835228bbc0cb99a695e81c45a2876a961af0c680662de4400ec96004c45bdc32726 68b01d7f861f2e03fc18adcc352bdeb7b06b39c2de749da45e9b48f46c45a390cb59 455dbf3e8be5938203def1c45af5eb5bf2037be005768d74faf11efd255b699f5ae9 bce8b3bf4688907d11ca89e7609b77c04abde845f63091b934017cf9a83f80eb86fd aa063630f86ee982d812062db0d1d760d84cfebbf99927cad516dd028bf2f9e9b7d2 4aa0bd9c0cb16a46c469eacfe1c48b8d7bf626e5d3eedb3f3cc4972f14a4ba605565 ec795dd9e68af4ac6185f370f1f77610c64b04d295007b34ed59ed7bceace7b9dea8 8346a2ee47dfec257ede5cd6985c34abaf5983d067786a695ae1039b631555bbe876 7baeedfe4f30df6b47479543c58a87e0aaf371234fc9094c4b2b79d025229634a3fd f68a36be0e8661a4f043504b7025f03f02e57704b4e410c60e69e88050d0c33ea71b c37f306c225f457fbab4929c65ba9401dc57a2dc31bd6d3f53099a00303132333435 363738393a3b3c3d3e3f input_share_1: > 000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f2021 22232425262728292a2b2c2d2e2f round_0: prep_share_0: > 7ae72bff1c063181bfea222dd77bcd40c5d15f62174d4af984d728effb82edd302 74c927a9ffa51b90c167c60b40c3cdda6f5f000d95651e628464f5d086fde6 prep_share_1: > 8718d400e3f9ce7e2415ddd2288432bf85623812cb8a7cdfed57f5a417c422e25e 84a67ab22c0b8ef7b0500c366feb95a544464ac776c3812e2299cce1d29095 prep_message: > 4f351791f4d706a31c54ce699ef1af97 out_share_0:  a18d0c7e2a74470727d4f628ff934e52 out_share_1:  c472f381d58bb8f8bc2b09d7006cb1ad agg_share_0: > a18d0c7e2a74470727d4f628ff934e52 agg_share_1: > c472f381d58bb8f8bc2b09d7006cb1ad agg_result: 100¶
buckets: [1, 10, 100] verify_key: "000102030405060708090a0b0c0d0e0f" upload_0: measurement: 50 nonce: "000102030405060708090a0b0c0d0e0f" public_share: > 5e015517900cfc204138c24f808ddf4ee85eca87ba246cd715d116195172e500 input_share_0: > c269f2189db431ca2ebe7230bacd692ac02a7790818681b6ee5ecbd509587b2976c7 897326840bb70083ee4df8ffe3dc65dbd71f56735a7dcc3a150da8e77171e6d380f2 f6886845e310dabec4382a037ffc3520737af8874f5b2aec2fae1405cce12c257ef2 9d22b3e511c30e8f9545251286b3ef3e3728256fc0cc21f5a8ea1d095187cc29132e 52d24ad778920c740af125833e9a8d2f6b0255740e8d2a4e7b718446610a0650ba40 61e6a3ee805c1dcffec90d00a3ea5817c9956459538c99efbb56b38eaa6aed93f5a0 0f11ab0210af030b91c56e6b2b47bb30fcdc5f44a28b5d733e4661cbd736ae071e78 4f99ba86ff2688f6751d2d16e80ddd99c9aceeb39ca660215f0fc1178f828e295840 f941fa9a3217afd52b32ec6c8ee3ec58291f2c88e8150374ccfd503f07bc0021d6a9 f1c81050df96ec5ce5c7d4f41255303132333435363738393a3b3c3d3e3f input_share_1: > 000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f2021 22232425262728292a2b2c2d2e2f round_0: prep_share_0: > ac872edaaaa532de61bcd63b168e522230b35064b09b6b35c1352bc354560fdeac 684da645d3e509c6db8ba31f58dabb5e015517900cfc204138c24f808ddf4e prep_share_1: > 5578d125555acd21824329c4e971adddc07187f0f25a30ceba8407513f60f8e3d1 5c17270c5631a718b3b4f5702aff54e85eca87ba246cd715d116195172e500 prep_message: > 890bfbdf0c619d5c853a92e28bce41b3 out_share_0:  c269f2189db431ca2ebe7230bacd692a  c02a7790818681b6ee5ecbd509587b29  76c7897326840bb70083ee4df8ffe3dc  65dbd71f56735a7dcc3a150da8e77171 out_share_1:  3f960de7624bce35b5418dcf453296d5  41d5886f7e797e49f5a0342af6a784d6  8c38768cd97bf448e37c11b207001c23  9c2428e0a98ca58217c5eaf257188e8e agg_share_0: > c269f2189db431ca2ebe7230bacd692ac02a7790818681b6ee5ecbd509587b2976c789 7326840bb70083ee4df8ffe3dc65dbd71f56735a7dcc3a150da8e77171 agg_share_1: > 3f960de7624bce35b5418dcf453296d541d5886f7e797e49f5a0342af6a784d68c3876 8cd97bf448e37c11b207001c239c2428e0a98ca58217c5eaf257188e8e agg_result: [0, 0, 1, 0]¶
verify_key: "000102030405060708090a0b0c0d0e0f" agg_param: (0, [0, 1]) upload_0: round_0: prep_share_0: > a2d369b7ee44e5f8b903e889b37ea7ef4d29ee36d1abdb13 prep_share_1: > 31182ff6e075b925a06c802eac3b14c1ad66ce991ca2ffc2 prep_message: > d2eb98add0ba9e1e587068b860babbb0fa8fbcd0ed4ddbd6 round_1: prep_share_0: > f4445265dc801ab5 prep_share_1: > 0dbbad9a227fe54a prep_message: > out_share_0:  d47170d4f804449e  e16add75c1d3f0c4 out_share_1:  2d8e8f2b06fbbb61  2195228a3d2c0f3b agg_share_0: > d47170d4f804449ee16add75c1d3f0c4 agg_share_1: > 2d8e8f2b06fbbb612195228a3d2c0f3b agg_result: [0, 1]¶
verify_key: "000102030405060708090a0b0c0d0e0f" agg_param: (1, [0, 1, 2, 3]) upload_0: round_0: prep_share_0: > 94352b588fbd978be7ef2cacb83b1b64c1d11af699253242 prep_share_1: > 4a3529a2b34159be1666064ba0a3a9cc4e72223ef6d0ac79 prep_message: > dd6a54fa43fff049fc5533f759dfc4300f443d3490f6debb round_1: prep_share_0: > 3dc2833b1f9010ee prep_share_1: > c43d7cc4df6fef11 prep_message: > out_share_0:  c03ede9261e7f441  8ef9fde3f50eb967  de3bb4acf0b9dff8  24687334644cc809 out_share_1:  41c1216d9d180bbe  7306021c09f14698  23c44b530e462007  de978ccb9ab337f6 agg_share_0: > c03ede9261e7f4418ef9fde3f50eb967de3bb4acf0b9dff824687334644cc809 agg_share_1: > 41c1216d9d180bbe7306021c09f1469823c44b530e462007de978ccb9ab337f6 agg_result: [0, 0, 0, 1]¶
verify_key: "000102030405060708090a0b0c0d0e0f" agg_param: (2, [0, 2, 4, 6]) upload_0: round_0: prep_share_0: > b640798e11bc37ac1f4981c99f0de15569b2bcc187d2f0f2 prep_share_1: > 37a720e1739caca1b39b8fe97ce96e09277eee9cc694c2cf prep_message: > ece7996f8658e44dd2e410b31cf74f5f8f30ab5e4f67b3c2 round_1: prep_share_0: > 52a1848f964385f8 prep_share_1: > af5e7b7068bc7a07 prep_message: > out_share_0:  352395a6c5e5804e  f9b61274892e76dd  b27e3eb4accbaa0a  3b1c4a7f78d59935 out_share_1:  ccdc6a59391a7fb1  0849ed8b75d18922  4f81c14b523455f5  c7e3b580862a66ca agg_share_0: > 352395a6c5e5804ef9b61274892e76ddb27e3eb4accbaa0a3b1c4a7f78d59935 agg_share_1: > ccdc6a59391a7fb10849ed8b75d189224f81c14b523455f5c7e3b580862a66ca agg_result: [0, 0, 0, 1]¶
verify_key: "000102030405060708090a0b0c0d0e0f" agg_param: (3, [1, 3, 5, 7, 9, 13, 15]) upload_0: round_0: prep_share_0: > 08efd12285f6e61599b2f9d941577fd843597d3b15f747d15b0b02daf6147752c6 a7a72723fa0cad98c7c38a0716ff8833dd7f97d40ed374f963dd2bced921614ccd 4dca993e6c4f5d1ab0ce9bf37df76b41e2c111868322dfc39b3145dfc832 prep_share_1: > 04eb21e562d79e4255b2c5001ed5949fc01841f40d921fdf5c1100b5c993f34e9d a143558cc507fb9c3e3832d85b751575b8f5a102257b141b94027bc0a849013f46 00d8257361b9d17e43f8cddc106b512f69c78b63bcde177e2d3c0cfb353d prep_message: > 1fdaf307e8cd8558ee64bfda5f2c14780472be2f238967b0b81c028fc0a86a2163 49eb7cafbf14a83506fcbcdf71749ea8957539d7334e8914f8dfa68e826b628b13 4ea2bfb1cd082f99f3c669d08e62bd704b899de93f01f741c96d51dafe6f round_1: prep_share_0: > 3e1ded312b78a4a2003d9365472d481c0adb6411d7049869693fb0b331750301 prep_share_1: > afe212ced4875b5dffc26c9ab8d2b7e3f5249bee28fb679696c04f4cce8afc7e prep_message: > out_share_0:  034a3226150cf3aa5e7f6e76b0c6a6b2aa9b750cdafdc6aae0ff2f4076894e2a  1413b483f800a4faa67b1e1c6bb1be2a1c9cc590761edb85f77692b67f9c696a  85591b91ee55008c7e7a58176d5bce3ca9974482b0c1aa2b7c69e0ab2e2e212c  03d9e6cde627cc30ceac7a6b5e3192a382d641e761c05c01eea4bc543c7fc41d  cd8891db6ac77a7fbba2c401c1488b8d31ee2cbc65cb26d02cb037cc99d32543  732e3e133c6d8c9a0f6647fc24be3c5d8a960b63c3163979fd4d42fd554a7812  8a2964a89cd70d7ba65bb52f0f15fb2db7dc1c534d743e37f223f74b8489c85d out_share_1:  eab5cdd9eaf30c55a18091894f39594d55648af3250239551f00d0bf8976b155  d9ec4b7c07ff5b055984e1e3944e41d5e3633a6f89e1247a08896d4980639615  68a6e46e11aaff738185a7e892a431c35668bb7d4f3e55d483961f54d1d1de53  ea26193219d833cf31538594a1ce6d5c7d29be189e3fa3fe115b43abc3803b62  20776e2495388580445d3bfe3eb77472ce11d3439a34d92fd34fc833662cda3c  7bd1c1ecc3927365f099b803db41c3a27569f49c3ce9c68602b2bd02aab5876d  63d69b576328f28459a44ad0f0ea04d24823e3acb28bc1c80ddc08b47b763722 agg_share_0: > 034a3226150cf3aa5e7f6e76b0c6a6b2aa9b750cdafdc6aae0ff2f4076894e2a1413b4 83f800a4faa67b1e1c6bb1be2a1c9cc590761edb85f77692b67f9c696a85591b91ee55 008c7e7a58176d5bce3ca9974482b0c1aa2b7c69e0ab2e2e212c03d9e6cde627cc30ce ac7a6b5e3192a382d641e761c05c01eea4bc543c7fc41dcd8891db6ac77a7fbba2c401 c1488b8d31ee2cbc65cb26d02cb037cc99d32543732e3e133c6d8c9a0f6647fc24be3c 5d8a960b63c3163979fd4d42fd554a78128a2964a89cd70d7ba65bb52f0f15fb2db7dc 1c534d743e37f223f74b8489c85d agg_share_1: > eab5cdd9eaf30c55a18091894f39594d55648af3250239551f00d0bf8976b155d9ec4b 7c07ff5b055984e1e3944e41d5e3633a6f89e1247a08896d498063961568a6e46e11aa ff738185a7e892a431c35668bb7d4f3e55d483961f54d1d1de53ea26193219d833cf31 538594a1ce6d5c7d29be189e3fa3fe115b43abc3803b6220776e2495388580445d3bfe 3eb77472ce11d3439a34d92fd34fc833662cda3c7bd1c1ecc3927365f099b803db41c3 a27569f49c3ce9c68602b2bd02aab5876d63d69b576328f28459a44ad0f0ea04d24823 e3acb28bc1c80ddc08b47b763722 agg_result: [0, 0, 0, 0, 0, 1, 0]¶