Verifiable Random Functions (VRFs)
Boston University
111 Cummington St, MCS135
Boston
MA
`02215`

USA
goldbe@cs.bu.edu
Boston University
111 Cummington St, MCS135
Boston
MA
`02215`

USA
reyzin@bu.edu
Hong Kong University of Science and Techology
Clearwater Bay
Hong Kong
dipapado@cse.ust.hkbu.edu
NS1
16 Beaver St
New York
NY
`10004`

USA
jvcelak@ns1.com
CFRG
public key cryptography
hashing
authenticated denial
A Verifiable Random Function (VRF) is the public-key version of a
keyed cryptographic hash. Only the holder of the private key
can compute the hash, but anyone with public key
can verify the correctness of the hash.
VRFs are useful for preventing enumeration of hash-based data structures.
This document specifies several VRF constructions that are secure in
the cryptographic random oracle model. One VRF uses RSA and the other
VRF uses Eliptic Curves (EC).
A Verifiable Random Function
(VRF) is the public-key version of a
keyed cryptographic hash. Only the holder of the private VRF key
can compute the hash, but anyone with corresponding public key
can verify the correctness of the hash.
A key application of the VRF is to provide privacy against
offline enumeration (e.g. dictionary attacks) on data stored in a
hash-based data structure.
In this application, a Prover holds the VRF private key and uses the VRF hashing to
construct a hash-based data structure on the input data.
Due to the nature of the VRF, only the Prover can answer queries
about whether or not some data is stored in the data structure. Anyone who
knows the public VRF key can verify that the Prover has answered the queries
correctly. However no offline inferences (i.e. inferences without querying
the Prover) can be made about the data stored in the data strucuture.
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in
.
The following terminology is used through this document:
The private key for the VRF.
The public key for the VRF.
The input to be hashed by the VRF.
The VRF hash output.
The VRF proof.
The Prover holds the private VRF key SK and public VRF key PK.
The Verifier holds the public VRF key PK.

A VRF comes with a key generation algorithm that generates a public VRF
key PK and private VRF key SK.
The prover hashes an input alpha using the private VRF key SK to obtain a VRF
hash output beta
beta = VRF_hash(SK, alpha)

The VRF_hash algorithm is deterministic, in
the sense that it always produces the same output beta given a
pair of inputs (SK, alpha).
The prover also uses the private key SK to construct a
proof pi that beta is the correct hash output
pi = VRF_prove(SK, alpha)

The VRFs defined in this document allow anyone to deterministically
obtain the VRF hash output beta directly from the proof value pi as
beta = VRF_proof2hash(pi)

Notice that this means that
VRF_hash(SK, alpha) = VRF_proof2hash(VRF_prove(SK, alpha))

and thus this document will specify VRF_prove and VRF_proof2hash
rather than VRF_hash.
The proof pi allows a Verifier holding the public key PK
to verify that beta is the correct VRF hash of input alpha
under key PK. Thus, the VRF also comes with an algorithm
VRF_verify(PK, alpha, pi)

that outputs VALID if beta=VRF_proof2hash(pi) is the correct
VRF hash of alpha under key PK, and outputs INVALID otherwise.
VRFs are designed to ensure the following security properties.
Uniqueness means that, for any fixed public
VRF key and for any input alpha, there is a unique VRF
output beta that can be proved to be valid. Uniqueness must hold
even for an adversarial Prover that knows the VRF private key SK.
More percisely, "full uniqueness" states that a computationally-bounded adversary cannot
choose
a VRF public key PK,
a VRF input alpha,
two different VRF hash outputs beta1 and beta2,
and two proofs pi1 and pi2 such that
VRF_verify(PK, alpha, pi1)
and VRF_verify(PK, alpha, pi2)
both output VALID.
A slightly weaker security
property called "trusted uniquness" sufficies for many applications.
Trusted uniqueness is the same as full uniqueness, but it must hold
only if the VRF keys PK and SK were generated in a trustworthy
manner. In other words, uniqueness might not hold if keys were
generated in an invalid manner or with bad randomness.
Like any cryprographic hash function, VRFs need to be
collision resistant. Collison resistance must hold
even for an adversarial Prover that knows the VRF private key SK.
More percisely, "full collision resistance" states that
it should be computationally
infeasible for an adversary to find two distinct VRF
inputs alpha1 and alpha2 that have the same VRF hash beta,
even if that adversary knows the private VRF key SK.
For most applications, a slightly weaker security property
called "trusted collision resistance" suffices.
Trusted collision resistance is the same as collision resistance,
but it holds only if PK and SK were generated in a trustworthy manner.
Pseudorandomness ensures that when an adversarial Verifier sees
a VRF hash output beta without its corresponding VRF proof pi,
then beta is indistinguishable from a random value.
More percisely, suppose the public and private VRF keys (PK, SK) were generated
in a trustworthy manner.
Pseudorandomness ensures that the VRF hash output beta
(without its corresponding VRF proof pi) on
any adversarially-chosen "target" VRF input alpha
looks indistinguishable from random
for any computationally bounded adversary who does not know the private
VRF key SK. This holds even if the adversary also gets to
choose other VRF inputs alpha' and observe their corresponding
VRF hash outputs beta' and proofs pi'.
With "full pseudorandomness", the adversary is allowed to choose the
"target" VRF input alpha at any time, even after it observes VRF outputs beta'
and proofs pi' on a variety of chosen inputs alpha'.
"Selective pseudorandomness" is a weaker security property
which suffices in many applications. Here, the adversary must choose
the target VRF input alpha independently of the public VRF key PK,
and before it observes VRF outputs beta'
and proofs pi' on inputs alpha' of its choice.
It is important to remember that the VRF output beta does not
look random to the Prover, or to any other party that knows the private
VRF key SK! Such a party can easily distinguish beta from
a random value by comparing beta to the result of VRF_hash(SK, alpha).
Also, the VRF output beta does not look random to any party that
knows valid VRF proof pi corresponding to the VRF input alpha, even
if this party does not know the private VRF key SK.
Such a party can easily distinguish beta from a random value by
checking whether VRF_verify(PK, alpha, pi) returns "VALID" and
beta = VRF_proof2hash(pi).
Also, the VRF output beta may not look random if VRF key generation
was not done in a trustworthy fashion. (For example, if VRF keys were
generated with bad randomness.)
[TODO: This property is not needed for applications
that use VRFs to prevent enumeration of hash-based data structures.
However, we noticed that some other applications of VRF
(e.g. Algorand, Oroborus) rely on
this property. We are waiting on a formal definition
of this property in the literature, and a proof that our ECVRF scheme
can satisfy this property.
Preliminary analysis suggests that acheiving this property
requires ECVRF verifiers to run an VRF_validate_key() key function upon receipt
of VRF public keys and the proof2hash function to be modified to take in
(gamma, beta, pk) rather than just gamma.]
Pseudorandomness, as defined in , does not
hold if the VRF keys were generated adversarially.
There is, however, a different type of pseudorandomness that could hold
even if the VRF keys are generated adversarially, as long as
the VRF input alpha is unpredictable. This property is similar
to the pseudorandomness achieved by an (ordinary, unkeyed)
cryptographic hash function.
[TODO: Formal definition here.]
The RSA Full Domain Hash VRF (RSA-FDH-VRF) is a VRF that satisfies
the "trusted uniqueness", "trusted
collision resistance", and "full pseudorandomness" properties defined in .
Its security follows from the
standard RSA assumption in the random oracle model. Formal
security proofs are in .
The VRF computes the proof pi as a deterministic RSA signature on
input alpha using the RSA Full Domain Hash Algorithm
parametrized with the selected hash algorithm.
RSA signature verification is used to verify the correctness of the
proof. The VRF hash output beta is simply obtained by hashing
the proof pi with the selected hash algorithm.
The key pair for RSA-FDH-VRF MUST be generated in a way that it satisfies
the conditions specified in Section 3 of .
In this document, the notation from is used.
Parameters used:
(n, e) - RSA public key
K - RSA private key
k - length in octets of the RSA modulus n

Fixed options:
Hash - cryptographic hash function
hLen - output length in octets of hash function Hash

Constraints on options:
Cryptographic security of Hash is at least as high as the
cryptographic security level of the RSA key.

Primitives used:
I2OSP - Coversion of a nonnegative integer to an octet string as defined in
Section 4.1 of
OS2IP - Coversion of an octet string to a nonnegative integer as defined in
Section 4.2 of
RSASP1 - RSA signature primitive as defined in
Section 5.2.1 of
RSAVP1 - RSA verification primitive as defined in
Section 5.2.2 of
MGF1 - Mask Generation Function based on a hash function as defined in
Section B.2.1 of

RSAFDHVRF_prove(K, alpha)
Input:
K - RSA private key
alpha - VRF hash input, an octet string

Output:
pi - proof, an octet string of length k

Steps:
EM = MGF1(alpha, k - 1)
m = OS2IP(EM)
s = RSASP1(K, m)
pi = I2OSP(s, k)
Output pi

RSAFDHVRF_proof2hash(pi)
Input:
pi - proof, an octet string of length k

Output:
beta - VRF hash output, an octet string of length hLen

Steps:
beta = Hash(pi)
Output beta

RSAFDHVRF_verify((n, e), alpha, pi)
Input:
(n, e) - RSA public key
alpha - VRF hash input, an octet string
pi - proof to be verified, an octet string of length n

Output:
"VALID" or "INVALID"

Steps:
s = OS2IP(pi)
m = RSAVP1((n, e), s)
EM = I2OSP(m, k - 1)
EM' = MGF1(alpha, k - 1)
If EM and EM' are equal, output "VALID";
else output "INVALID".

The Elliptic Curve Verifiable Random Function (ECVRF) is a VRF that
satisfies the trusted uniqueness, trusted collision resistance,
and full pseudorandomness properties defined in .
The security of this VRF follows from the decisional
Diffie-Hellman (DDH) assumption in the random oracle model. Formal security proofs are
in .
To additionally satisfy "full uniqueness" and "full collision resistance",
the Verifier MUST additionally perform the validation procedure specified in
upon receipt of the public
VRF key.
Fixed options (specified in ):
F - finite field
2n - length, in octets, of a field element in F; must be even
E - elliptic curve (EC) defined over F
m - length, in octets, of an EC point encoded as an octet string
G - subgroup of E of large prime order
q - prime order of group G; must be less than 2^{2n}
cofactor - number of points on E divided by q
g - generator of group G
Hash - cryptographic hash function
hLen - output length in octets of Hash; must be at least 2n
suite - a single nonzero octet specifying the ECVRF ciphersuite, which determines the above options

Notation and primitives used:
p^k - When p is an EC point: point multiplication, i.e. k repetitions of the EC group operation applied to the EC point p. When p is an integer: exponentiation
|| - octet string concatenation
I2OSP - nonnegative integer conversion to octet string as defined in
Section 4.1 of
OS2IP - Coversion of an octet string to a nonnegative integer as defined in
Section 4.2 of
EC2OSP - conversion of EC point to an m-octet string
as specified in
OS2ECP - conversion of an m-octet string to EC point
as specified in .
OS2ECP returns INVALID if the octet string does not convert to a valid EC point.
RS2ECP - conversion of a random 2n-octet string to an
EC point as specified in
ECVRF_hash_to_curve - collision resistant hash of strings
to an EC point; options described in and specified in .
ECVRF_nonce_generation - derives a pseudorandom nonce
from SK and the input as part of ECVRF proving.
Specified in
ECVRF_hash_points - collision resistant hash of EC points
to an n-octet string. Specified in .

Parameters used (the generation of these parameters is specified in ):
SK - VRF private key
x - VRF secret scalar, an integer
Note: depending on the ciphersuite used, the VRF secret scalar may be equal
to SK; else, it is derived from SK

y = g^x - VRF public key, an EC point

Note: this function must have the VRF private key SK as input.
Below we make it more
efficient by supplying it also with the secret scalar x and the public key y as
additional inputs; however, each of these can be computed from SK if desired.
ECVRF_prove(y, x, alpha)
Input:
SK - VRF private key
x - VRF secret scalar
y = g^x - VRF public key

Output:
pi - VRF proof, octet string of length m+3n

Steps:
h = ECVRF_hash_to_curve(suite, y, alpha)
h1 = EC2OSP(h)
gamma = h^x
k = ECVRF_nonce_generation(SK, h1)
c = ECVRF_hash_points(h, gamma, g^k, h^k)
s = (k + c*x) mod q (where * denotes integer multiplication)
pi = EC2OSP(gamma) || I2OSP(c, n) || I2OSP(s, 2n)
Output pi

ECVRF_proof2hash(pi)
Input:
pi - VRF proof, octet string of length m+3n

Output:
"INVALID", or
beta - VRF hash output, octet string of length 2n

Steps:
D = ECVRF_decode_proof(pi)
If D is "INVALID", output "INVALID" and stop
(gamma, c, s) = D
three = 0x03 = I2OSP(3, 1), a single octet with value 3
preBeta = Hash(suite || three || EC2OSP(gamma^cofactor))
beta = first 2n octets of preBeta
Output beta

ECVRF_verify(y, pi, alpha)
Input:
y - public key, an EC point
pi - VRF proof, octet string of length 5n+1
alpha - VRF input, octet string

Output:
"VALID" or "INVALID"

Steps:
D = ECVRF_decode_proof(pi)
If D is "INVALID", output "INVALID" and stop
(gamma, c, s) = D
u = g^s / y^c (where / denotes EC point subtraction, i.e. the group operation applied to g^s and the inverse of y^c)
h = ECVRF_hash_to_curve(suite, y, alpha)
v = h^s / gamma^c (where / again denotes EC point subtraction)
c' = ECVRF_hash_points(h, gamma, u, v)
If c and c' are equal, output "VALID";
else output "INVALID"

The ECVRF_hash_to_curve algorithm takes in the VRF input alpha
and converts it to h, an EC point in G.
This algorithm is the only place alpha is used in the proving and verfying. See
for further discussion.
The algorithms in this section are not compatible with each other; the choice of algorithm is made in .
The following ECVRF_hash_to_curve_try_and_increment(suite, y, alpha) algorithm
implements ECVRF_hash_to_curve in a simple and
generic way that works for any elliptic curve.
The running time of this algorithm depends on alpha.
For the ciphersuites specified
in , this algorithm
is expected to find a valid curve point after approximately two attempts
(i.e., when ctr=1) on average. See also .
However, because the running time of algorithm depends on alpha,
this algorithm SHOULD be avoided in
applications where it is important that
the VRF input alpha remain secret.
ECVRF_hash_to_try_and_increment(suite, y, alpha)
Input:
suite - a single octet specifying ECVRF ciphersuite.
y - public key, an EC point
alpha - value to be hashed, an octet string

Output:
h - hashed value, a finite EC point in G

Steps:
ctr = 0
pk = EC2OSP(y)
one = 0x01 = I2OSP(1, 1), a single octet with value 1
h = "INVALID"
While h is "INVALID" or h is EC point at infinity:
CTR = I2OSP(ctr, 4)
ctr = ctr + 1
hash = Hash(suite || one || pk || alpha || CTR)
attempted_hash = first 2n bits of hash
h = RS2ECP(attempted_hash)
If h is not "INVALID" and cofactor > 1, set h = h^cofactor

Output h

The following ECVRF__hash_to_curve_elligator2_25519(suite, y, alpha)
algorithm implements ECVRF_hash_to_curve using the elligator2
algorithm exclusively for Curve25519.
ECVRF_hash_to_curve_elligator2_25519(y, alpha)
Input:
alpha - value to be hashed, an octet string
y - public key, an EC point
suite = a single octet specifying ECVRF ciphersuite.

Output:
h - hashed value, a finite EC point in G
per the encoding in RFC8032 Section 5.1.2

Fixed options:
p = 2^255-19, the size of the finite field F, a prime, for Curve25519
A = 486662, Montgomery curve constant for Curve25519
cofactor = 8 , the cofactor for Curve25519
* denotes integer multiplication

Constraints on options:
output length of Hash is at least 16n + 1 bits
(i.e. greater than 2n octets)

Steps:
pk = EC2OSP(y)
one = 0x01 = I2OSP(1, 1), a single octet with value 1
hash = Hash(suite || one || pk || alpha )
r = first 2n octets of hash
x_0 = next unused bit of hash
u = - A / (1 + 2*(r^2) ) mod p
v = u * (u^2 + A*u + 1) mod p
Let e equal the Legendre symbol of v modulo p
If e is equal to 1 then finalu = u; else finalu = (-A - u) mod p
y = (finalu - 1) / (finalu + 1) mod p
let h = (x_0,y), an EC point per the encoding in Section 5.1.2 of
h = h^cofactor

In order to make this algorithm run in time that is (almost) independent
of the input (so-called "constant-time"), implementers should pay particular attention
to Steps 7 and 8 above.
These steps can be implemented using the following approach:
e = v ^ ((p-1)/2) mod p
finalu = (e*u + (e-1) * (A/2)) mod p

The first step will produce a value e that is either 1 or p-1. The second step is so fast compared
to the first that that its dependence on e is likely to not matter; implementers can also ensure that it
runs in the same amount of time regardless of e.
If having this algorithm run in constant time is not important, then there are much faster algorithms
to compute the Legendre symbol (which is the same as the Jacobi symbol because p is a prime).
See, for example, Section 12.3 of .
[TODO: In addition to Elligator,
it would be nice to have a generic ECVRF_hash_to_curve algorithm that
runs in constant time and
provides a generic way to hash an octet string onto any elliptic curve.
There is an an upcoming draft that deals with this
.
Some other useful algorithms include , and
Shallue-Woestijne-Ulas algorithm from .]
ECVRF_hash_points(p_1, p_2, ..., p_j)
Input:
p_i - EC point in G

Output:
c - hash value, integer between 0 and 2^(8n)-1

Steps:
two = 0x02 = I2OSP(2, 1), a single octet with value 2
Initialize str = suite || two
for p_i in [p_1, p_2, ... p_j]:
str = str || EC2OSP(p_i)
c1 = Hash(str)
c2 = first n octets of c1
c = OS2IP(c2)
Output c

ECVRF_decode_proof(pi)
Input:
pi - VRF proof, octet string (m+3n octets)

Output:
"INVALID", or
gamma - EC point
c - integer between 0 and 2^(8n)-1
s - integer between 0 and 2^(16n)-1

Steps:
let gamma', c', s' be pi split after m-th and m+n-th octet
gamma = OS2ECP(gamma')
if gamma = "INVALID" output "INVALID" and stop.
c = OS2IP(c')
s = OS2IP(s')
Output gamma, c, and s

This document defines ECVRF-P256-SHA256 as follows:
suite = 0x01 = I2OSP(1, 1).
The EC group G is the NIST-P256 elliptic curve, with curve parameters
as specified in (Section D.1.2.3)
and (Section 2.6). For this group,
2n = 32 and cofactor = 1.
The key pair generation primitive is specified in
Section 3.2.1 of (q, g, SK, and PK in this document
correspond to in n, G, d, and Q in Section 3.2.1 of ).
In this ciphersuite, the secret scalar x is equal to the private key SK.
The ECVRF_nonce_generation function is as specified in Section 3.2 Steps b-h
(omitting Step a and using h1 that is provided as input to ECVRF_nonce_generation), with the hash function
SHA-256 as specified in
EC2OSP is specified in Section 2.3.3 of with point compression on.
This implies m = 2n + 1 = 33.
OS2ECP is specified in Section 2.3.4 of .
RS2ECP(h) = OS2ECP(0x02 || h) (where 0x02 is a single octet with value 2, 0x02=I2OSP(2, 1)). The input h is a 32-octet string
and the output is either an EC point or "INVALID".
The hash function Hash is SHA-256 as specified in .
The ECVRF_hash_to_curve function is as specified in .

This document defines ECVRF-ED25519-SHA512 as follows:
suite = 0x02 = I2OSP(2, 1).
The EC group G is the Ed25519
elliptic curve with parameters defined in Table 1 of
.
For this group, 2n = 32 and cofactor = 8.
The private key and generation of the secret scalar and the public
key are specified in Section 5.1.5 of
The ECVRF_nonce_generation function is as specified in
Section 5.1.6 Steps 1-2, with dom2(F, C) equal to the empty string and PH(M) equal to h1.
EC2OSP is specified in Section 5.1.2 of .
This implies m = 2n = 32.
OS2ECP is specified in Section 5.1.3 of .
RS2ECP is equivalent to OS2ECP.
The hash function Hash is SHA-512 as specified in .
The ECVRF_hash_to_curve function is as specified in .

This document defines ECVRF-ED25519-SHA512-Elligator2 as follows:
This ciphersuite is identical to ECVRF-ED25519-SHA512 except that
the ECVRF_hash_to_curve function is as specified in and
suite = 0x03 = I2OSP(3, 1).

The ECVRF as specified above is a VRF that satisfies the
"trusted uniqueness", "trusted collision resistance", and
"full pseudorandomness" properties defined in .
In order to obtain "full uniqueness" and "full collision resistance" (which provide
protection against a malicious VRF public key), the Verifier MUST
perform
the following additional validation procedure upon receipt of the public
VRF key. The public VRF key MUST NOT be used if this procedure returns "INVALID".
Note that this procedure is not sufficient if the elliptic curve E
or the point g, the generator of group G, is untrusted. If the prover is untrusted,
the Verifier MUST
obtain E and g from a trusted source, such as a ciphersuite specification, rather
than from the prover.
This procedure supposes that the public key provided to the Verifier is an octet
string. The procedure returns "INVALID" if the public key in invalid.
Otherwise, it returns y, the public key as an EC point.
ECVRF_validate_key(PK)
Input:
PK - public key, an octet string

Output:
"INVALID", or
y - public key, an EC point

Steps:
y = OS2ECP(PK)
If y is "INVALID", output "INVALID" and stop
If y^cofactor is the EC point at infinty, output "INVALID" and stop
Output y

An implementation of the RSA-FDH-VRF (SHA-256) and ECVRF-P256-SHA256 was
first developed
as a part of the NSEC5 project and is available
at .
The ECVRF implementation may be out of date as this spec has evolved.
The Key Transparency project at Google
uses a VRF implemention that is similar to
the ECVRF-P256-SHA256, with a few minor changes
including the use of SHA-512 instead of SHA-256. Its implementation
is available
An implementation by Yahoo! similar to the ECVRF is available at
.
An implementation similar to ECVRF is available as part of the
CONIKS implementation in Golang at
.
Open Whisper Systems also uses a VRF very similar to
ECVRF-ED25519-SHA512-Elligator, called VXEdDSA, and specified here:
Applications that use the VRFs defined in this
document MUST ensure that that the VRF key is generated correctly,
using good randomness.
The ECVRF as specified in -
statisfies the "trusted uniqueness" and "trusted collision resistance" properties
as long as the VRF keys are generated correctly, with good randomness.
If the Verifier trusts the VRF keys are generated correctly, it MAY use
the public key y as is.
However, if the ECVRF uses keys that could be generated adversarially, then the
the Verfier MUST first perform the validation procedure ECVRF_validate_key(PK)
(specified in ) upon receipt of the
public key PK as an octet string. If the validation procedure
outputs "INVALID", then the public key MUST not be used.
Otherwise, the procedure will output a valid public key y,
and the ECVRF with public key y satisfies the "full uniqueness" and
"full collision resistance" properties.
The RSA-FDH-VRF statisfies the "trusted uniqueness" and "trusted collision resistance" properties
as long as the VRF keys are generated correctly, with good randomness.
These properties may not hold if the keys are generated adversarially
(e.g., if RSA is not permutation). Meanwhile,
the "full uniqueness" and "full collision resistance" are
properties that hold even if VRF keys are generated by an adversary.
The RSA-FDH-VRF defined in this document does not have these properties.
However, if adversarial key generation is a concern, the
RSA-FDH-VRF may be modifed to have these
properties by adding additional cryptographic checks
that its public key has the right form. These modifications are left for future specification.
Without good randomness, the "pseudorandomness"
properties of the VRF may not hold. Note that it is not possible to guarantee
pseudorandomness in the face of adversarially generated VRF keys. This is
because an adversary can always use bad randomness to generate the VRF keys,
and thus, the VRF output may not be pseudorandom.
presents cryptographic reductions to an
underlying hard problem (e.g. Decisional Diffie Hellman for the ECVRF, or the
standard RSA assumption for RSA-FDH-VRF) that prove the VRFs specificied in this
document possess full pseudorandomness
as well as selective pseudorandomness.
However, the cryptographic reductions are tighter for selective
pseudorandomness than for full pseudorandomness. This means the
the VRFs have quantitavely stronger security
guarentees for selective pseudorandomness.
Applications that are concerned about tightness of cryptographic
reductions therefore have two options.
They may choose to ensure that selective pseudorandomness is sufficient for
the application. That is, that
pseudorandomness of outputs matters only for inputs that are chosen
independently of the VRF key.
If full pseudorandomness is required for the application, the application
may increase
security parameters to make up for the loose security reduction.
For RSA-FDH-VRF, this means increasing the RSA key length. For
ECVRF, this means increasing the cryptographic strength of the EC group
G. For both RSA-FDH-VRF and ECVRF the cryptographic strength of the
hash function Hash may also potentially need to be increased.

The security of the ECVRF defined in this document relies on the
fact that nonce k used in the ECVRF_prove algorithm is
chosen uniformly and pseudorandomly modulo q, and is unknown to the advesrary.
Otherwise, an adversary may be able to recover
the private VRF key x (and thus break pseudorandomness of the VRF)
after observing several valid VRF proofs pi. The nonce generation methods
specified in the ECVRF ciphersuites of
are designed with this requirement in mind.
The ECVRF_hash_to_curve_try_and_increment algorithm defined in
SHOULD NOT be used in applications where
the VRF input alpha is secret and is hashed by the VRF on-the-fly.
This is because the algorithm's running time depends
on the VRF input alpha, and thus creates a timing channel that
can be used to learn information about alpha.
That said, for most inputs the amount of information obtained from
such a timing attack is likely to be small (1 bit, on average), since the algorithm
is expected to find a valid curve point after only two attempts.
However, there might be inputs which cause the algorithm to make many attempts
before it finds a valid curve point; for such inputs, the information leaked
in a timing attack will be more than 1 bit.
Meanwhile, ECVRF-ED25519-SHA512-Elligator2 runs in constant time
if the implementation of the ECVRF_hash_to_curve function
specified in also runs in constant time.
The VRF proof pi is not designed to provide secrecy and, in general,
may reveal the VRF input alpha.
Anyone who knows PK and pi is able to perform an offline
dictionary attack to search for alpha, by verifying guesses for alpha using VRF_verify.
This is in contrast to the VRF hash output beta which, without the proof, is pseudorandom
and thus is designed to reveal no information about alpha.
The VRFs specified in this document allow for read-once access to
the input alpha for both signing and verifying. Thus, additional
prehashing of alpha (as specified, for example, in
for EdDSA signatures) is not needed,
even for applications that need to handle long alpha or
to support the
Initialized-Update-Finalize (IUF) interface (in such an interface,
alpha is not supplied
all at once, but rather in pieces by a sequence of calls to Update).
The ECVRF, in particular, uses alpha only in
ECVRF_hash_to_curve. The curve point h becomes the representative
of alpha thereafter. Note that the suite octet and the public key
are hashed together with alpha in ECVRF_hash_to_curve, which ensures
that the curve (including the generator g) and the public
key are included indirectly into subsequent hashes.
Hashing is used for four different purposes in ECVRF (namely, in hash_to_curve, nonce_generation, hash_points, and proof2hash). The
theoretical analysis assumes each of these four functions is a separate random oracle.
This analysis still holds even if the same hash function is used, as long as the four
queries made to the hash function for a given SK and alpha are overwhelmingly unlikely
to equal each other or to any queries made to the hash function for the same SK and
different alpha. This is indeed the case for the ECVRF ciphersuites defined in this document, because:
inputs to the hash function used during nonce_generation are unlikely to equal
to inputs given to hash_to_curve, proof2hash, and hash_points. This
follows since nonce_generation inputs a secret to the hash function that is not used by
honest parties as input to any other hash function, and is not available to the adversary
the second octet of the input to the hash function used in
hash_to_curve, proof2hash, and
hash_points are all different

If future designs need to specify variants (e.g., additional ciphersuites) of the design in this document,
then, to avoid the possibility
that an adversary can obtain a VRF output under one variant, and then claim it was obtained under
another variant,
they should specify a different suite constant. This way, the inputs
to the hash_to_curve hash function used in producing h are
guaranteed to be different; since all the other hashing done by the prover
depends on h, inputs all the hash functions used by the prover will also be
different as long as hash_to_curve is collision resistant.
Note to RFC Editor: if this document does not obsolete an existing RFC,
please remove this appendix before publication as an RFC.
00 - Forked this document from draft-goldbe-vrf-01.
01 - Minor updates, mostly highlighting TODO items.
02 - Added specification of elligator2 for Curve25519, along
with ciphersuites for ECVRF-ED25519-SHA512-Elligator.
Changed
ECVRF-ED25519-SHA256 suite to ECVRF-ED25519-SHA512. (This change
made because Ed25519 in signatures
use SHA512 and not SHA256.)
Made ECVRF nonce generation a separate component, so that nonces are determinsitic.
In ECVRF proving, changed + to - (and made corresponding
verification changes) in order to be consistent with EdDSA and ECDSA.
Highlighted that ECVRF_hash_to_curve acts like a prehash.
Added "suites" variable to ECVRF for future-proofing.
Ensured domain separation for hash functions by modifying hash_points and added
discussion about domain separation.
Updated todos in the "additional pseudorandomness property"
section. Added an discussion of secrecy into security considerations.
Removed g and PK=y from ECVRF_hash_points because they are already present
via h, which is computed via hash_to_curve using the suite (which identifies g) and y.

This document also would not be possible without the work of
Moni Naor (Weizmann Institute),
Sachin Vasant (Cisco Systems), and
Asaf Ziv (Facebook).
Shumon Huque (Salesforce), David C. Lawerence (Akamai), and Trevor Perrin provided
valuable input to this draft.
Digital Signature Standard (DSS)
National Institute for Standards and Technology
SEC 1: Elliptic Curve Cryptography
Standards for Efficient Cryptography Group (SECG)
A Computational Introduction to Number Theory and Algebra
Making NSEC5 Practical for DNSSEC
Verifiable Random Functions
How to Hash into Elliptic Curves
Efficient Indifferentiable Hashing into Ordinary Elliptic Curves
Hashing to Elliptic Curves