Verifiable Random Functions (VRFs)
Boston University
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goldbe@cs.bu.edu
University of Maryland
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dipapado@bu.edu
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jvcelak@ns1.com
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.
The main application of the VRF is to protect the privacy of data records
stored in a hash-based data structure against a querying adversary. In this
application, a prover holds the VRF secret key and uses the VRF hashing to
construct a hash-based data structure on the input data.
Due to the nature of the VRF hashing, 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.

A VRF comes with a key generation algorithm that generates a public VRF
key PK and private VRF key SK.
A VRF 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 MUST be deterministic, in
the sense that it will always produce the same output beta given a
pair of inputs (SK, alpha).
The private key SK is also used 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))

The proof pi allows anyone 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 correct
VRF hash of alpha under key PK, and outputs INVALID otherwise.
VRFs are designed to ensure the following security properties.
Uniqueness states 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, even for a
computationally-bounded adversary that knows the VRF secret key SK.
More precisely, full uniqueness states that
a computationally bounded adversary cannot
choose
a VRF public key PK,
a VR 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 otherwords, uniqueness might not hold if keys were
generated in an invalid manner.
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 a party that knows the private VRF key SK! Such a party
can easily distinguish beta from a random value by comparing it 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).
Finally, 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
not generated with good randomness.)
Finally, like any cryprographic hash function, VRFs need to be
collision resistant. Specifically, 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 secret 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.
The RSA Full Domain Hash VRF (RSA-FDH-VRF) is VRF that satisfies
the trusted uniqueness, full pseudorandomness, and trusted
collision resistance 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.
Used parameters:
(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

Options constraints:
Cryptographic security of Hash is at least as high as the cryptographic security level of the RSA key

Used primitives:
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 the same, output "VALID";
else output "INVALID".

The Elliptic Curve Verifiable Random Function (EC-VRF) is VRF that
satisfies the trusted uniqueness, full pseudorandomness, and
trusted collision resistance properties defined in .
The security of this VRF
follows from the decisional Diffie-Hellman (DDH) assumption in the
cyclic group in the random oracle model. Formal security proofs are
in .
The key pair generation primitive is specified in
Section 3.2.1 of .
Fixed options:
G - EC group
q - prime order of group G
g - generator of group G
2n - ceil(log2(q)/8); where log2(x) is the binary
logarithm of x and ceil(x) is the smallest integer
larger than or equal to the real number x.
Hash - cryptographic hash function
hLen - output length in octets of function Hash

Options constraints:
Cryptographic security of Hash is at least as high as the cryptographic security of G
hLen is equal to 2n

Used parameters:
g^x - EC public key
x - EC private key

Used primitives:
"" - empty octet string
|| - octet string concatenation
p^k - EC point multiplication
p1*p2 - EC point addition
h[i] - the i'th octet of octet string h
ECP2OS - EC point to octet string conversion with point compression as specified in Section 2.3.3 of
OS2ECP - octet string to EC point conversion with point compression as specified in Section 2.3.4 of

ECVRF_prove(g^x, x, alpha)
Input:
g^x - EC public key
x - EC private key
alpha - VRF input, octet string

Output:
pi - VRF proof, octet string of length 5n+1

Steps:
h = ECVRF_hash_to_curve(alpha, g^x)
gamma = h^x
choose a random nonce k from [0, q-1]
c = ECVRF_hash_points(g, h, g^x, h^x, g^k, h^k)
s = k - c*q mod q
pi = ECP2OS(gamma) || I2OSP(c, n) || I2OSP(s, 2n)
Output pi

ECVRF_proof2hash(pi)
Input:
pi - VRF proof, octet string of length 5n+1

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

Steps:
beta = pi[2] || pi[3] || ... pi[2n+1]
Output beta

ECVRF_verify(g^x, pi, alpha)
Input:
g^x - EC public key
pi - VRF proof, octet string of length 5n+1
alpha - VRF input, octet string

Output:
"VALID" or "INVALID"

Steps:
gamma, c, s = ECVRF_decode_proof(pi)
If gamma is not a valid EC point in G,
output "INVALID" and stop.
u = (g^x)^c * g^s
h = ECVRF_hash_to_curve(alpha, g^x)
v = gamma^c * h^s
c' = ECVRF_hash_points(g, h, g^x, gamma, u, v)
If c and c' are the same, output "VALID";
else output "INVALID".

The ECVRF_hash_to_curve algorithm takes in an octet string alpha
and converts it to h, an EC point in G.
The following ECVRF_hash_to_curve1(alpha, g^x) algorithm
implements ECVRF_hash_to_curve in a simple and
generic way that works for any elliptic curve that supports
point compression.
However, this algorithm MUST NOT be used in
applications where
the VRF input alpha must be kept secret. This is because the running time of
the hashing algorithm depends on alpha, and so it is susceptible to
timing attacks. That said, the amount of information obtained from
such a timing attack is likely to be small, since the algorithm
is expected to find a valid curve point after only two attempts
(i.e., when ctr=1) on average (see ).
ECVRF_hash_to_curve1(alpha, g^x)
Input:
alpha - value to be hashed, an octet string
g^x - EC public key

Output:
h - hashed value, EC point in G

Steps:
ctr = 0
pk = ECP2OS(g^x)
Repeat:
CTR = I2OSP(ctr, 4)
p = 0x02 || Hash(alpha || pk || CTR)
Goto step 3 if OS2ECP(p) is valid EC point in G
p = 0x03 || Hash(alpha || pk || CTR)
Goto step 3 if OS2ECP(p) is valid EC point in G
ctr = ctr + 1

h = OS2ECP(p)
Output h

The initial octet 0x02 in the octet string created in step B
represents that the point in compressed form has positive
y-coefficient . Similarly, the 0x03 octet
in step D represents negative y-coefficient.
For applications where VRF input alpha must be kept secret,
the following ECVRF_hash_to_curve algorithm MAY be used to used as
generic way to hash an octet string onto any elliptic curve.
[TODO: If there interest, we could look into specifying the
generic deterministic time hash_to_curve algorithm from
. ]
ECVRF_hash_points(p_1, p_2, ..., p_j)
Input:
p_i - EC point in G

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

Steps:
P = ""
for p_i in [p_1, p_2, ... p_j]:
P = P || ECP2OS(p_i)
h' = Hash(P)
h = OS2IP(h'[1] || h'[2] || ... h'[n])
Output h

ECVRF_decode_proof(pi)
Input:
pi - VRF proof, octet string (5n+1 octets)

Output:
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 (2n+1)-th and (3n+1)-th octet
gamma = OS2ECP(gamma')
c = OS2IP(c')
s = OS2IP(s')
Output gamma, c, and s

[Seeking feedback on this section!]
This document defines EC-VRF-P256-SHA256 as follows:
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,
the length in octets of a single coordinate of an EC point is
2n = 32.
The hash function Hash is SHA-256 as specified in .
The ECVRF_hash_to_curve function is ECVRF_hash_to_curve1,
as specified in .

This document defines EC-VRF-ED25519-SHA256 as follows:
The EC group G is the Ed25519
elliptic curve with parameters defined in
(Section 4.1).
For this group,
the length in octets of a single coordinate of an EC point is
2n = 32.
The hash function Hash is SHA-256 as specified in .
The ECVRF_hash_to_curve function is as specified in .

[TODO: Should we add an EC-VRF-ED25519-SHA256-Elligator
ciphersuite where the Elligator hash function is used for ECVRF_hash-to-curve?]
[TODO: Add an Ed448 ciphersuite?]
An implementation of the RSA-FDH-VRF (SHA-256) and EC-VRF-P256-SHA256 was developed
as a part of the NSEC5 project and is available
at .
The Key Transparency project at Google
uses a VRF implemention that is almost identical to
the EC-VRF-P256-SHA256 specified here, with a few minor changes
including the use of SHA-512 instead of SHA-256. Its implementation
is available
Open Whisper Systems also uses a VRF very similar to
EC-VRF-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. Without good randomness, pseudorandomness
properties of the VRF may not hold. Also, trusted uniqueness and trusted collision-resistance
may also not hold if the keys are generated
adversarially (e.g., the RSA modulus is not a product of two primes
for the RSA-FDH-VRF or the public key g^x is not valid point in the
prime-order group G for the EC).
Full uniqueness and full collision-resistance (as opposed to trusted
uniqueness and trusted collision-resistance) are properties that hold
even if VRF keys are generated by an adversary.
The VRFs defined in this document
do not have these properties.
However, they may be modifed to have these
properties if adversarial key generation
is a concern. The modification consists of additional cryptographic
proofs that keys
have of the correct form.
These modifications are left for future specification.
Note that for the RSA-FDH-VRF, it might be possible to construct
such a proof using the
identification protocol made non-interactive using the Fiat-Shamir heuristic
in the random oracle model.
However, 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.
Applications that use the EC-VRF defined in this document MUST ensure
that the random nonce k used in the ECVRF_prove algorithm is
chosen with proper randomness. 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 EC-VRF_hash_to_curve 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 EC-VRF_hash_to_curve 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.
presents cryptographic reductions to an
underlying hard problem (e.g. Decisional Diffie Hellman, or the
standard RSA assumption) 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 therefor 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
EC-VRF, this means increasing the cryptographic strength of the EC group
G. For both RSA-FDH-VRF and EC-VRF the cryptographic strength of the
hash function Hash may also potentially need to be increased.

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-vcelak-nsec5-04.
Cleaned up the definitions of VRF algorithms.
Added security definitions for VRF and security considerations.
Parameterized EC-VRF so it could support curves other than
P-256 and Ed25519.

Leonid Reyzin (Boston University) made major contributions to this
document.
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) and David C. Lawerence (Akamai) 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)
NSEC5 from Elliptic Curves
Verifiable Random Functions
Wallet databases with observers
How to Hash into Elliptic Curves
A Practical Zero-Knowledge Protocol Fitted to Security Microprocessor Minimizing Both Transmission and Memory
Note to RFC Editor: please remove this appendix before publication as an RFC.
Open issues