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_proof_to_hash(pi)

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

and thus this document will specify VRF_prove and VRF_proof_to_hash
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, beta = VRF_proof_to_hash(pi)) if pi is valid,
and 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 precisely, "full uniqueness" states that a computationally-bounded adversary cannot
choose
a VRF public key PK,
a VRF input alpha,
and two proofs pi1 and pi2 such that
VRF_verify(PK, alpha, pi1) outputs (VALID, beta1),
VRF_verify(PK, alpha, pi2) outputs (VALID, beta2),
and beta1 is not equal to beta2.
A slightly weaker security
property called "trusted uniqueness" 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 precisely, "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 precisely, 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, beta).
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.)
Pseudorandomness, as defined in , does not
hold if the VRF keys were generated adversarially. For instance, if an adversary outputs VRF keys that are deterministically generated (or hard-coded and publicly known), then the outputs are easily derived by anyone.
There is, however, a different type of unpredictability that is desirable in certain VRF applications (such as and ). This property is similar
to the unpredictability achieved by an (ordinary, unkeyed)
cryptographic hash function: if the input has enough entropy (i.e., cannot be predicted), then the correct output is indistinguishable
from uniform.
Although neither formal definitions nor proofs of this property have appeared in cryptographic literature, the VRF schemes presented in this specification are believed to satisfy this property if the public key was generated in a trustworthy
manner. Additionally, the ECVRF also satisifies this property even if the public key was not generated in a trustworthy manner, as long as
the public key satisfies the key validation
procedure in .
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 (k must be less than 2^32)

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

Primitives used:
I2OSP - Conversion of a nonnegative integer to an octet string as defined in
Section 4.1 of
OS2IP - Conversion 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 the hash function Hash as defined in
Section B.2.1 of
|| - octet string concatenation

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

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

Steps:
one_string = 0x01 = I2OSP(1, 1), a single octet with value 1
EM = MGF1(one_string || I2OSP(k, 4) || I2OSP(n, k) || alpha_string, k - 1)
m = OS2IP(EM)
s = RSASP1(K, m)
pi_string = I2OSP(s, k)
Output pi_string

RSAFDHVRF_proof_to_hash(pi_string)
Input:
pi_string - proof, an octet string of length k

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

Important note:
RSAFDHVRF_proof_to_hash should be run only on pi_string that is known to have been produced by RSAFDHVRF_prove, or from within RSAFDHVRF_verify as specified in .

Steps:
two_string = 0x02 = I2OSP(2, 1), a single octet with value 2
beta_string = Hash(two_string || pi_string)
Output beta_string

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

Output:
("VALID", beta_string), where beta_string is the VRF hash output, an octet string of length hLen; or
"INVALID"

Steps:
s = OS2IP(pi_string)
m = RSAVP1((n, e), s)
EM = I2OSP(m, k - 1)
one_string = 0x01 = I2OSP(1, 1), a single octet with value 1
EM' = MGF1(one_string || I2OSP(k, 4) || I2OSP(n, k) || alpha_string, k - 1)
If EM and EM' are equal, output ("VALID", RSAFDHVRF_proof_to_hash(pi_string));
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, rounded up to the nearest even integer
E - elliptic curve (EC) defined over F
ptLen - 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
qLen - length of q in octets, i.e., smallest integer such that 2^(8qLen)>q (note that in the typical case, qLen equals 2n or is close to 2n)
cofactor - number of points on E divided by q
B - generator of group G
Hash - cryptographic hash function
hLen - output length in octets of Hash; must be at least 2n
suite_string - a single nonzero octet specifying the ECVRF
ciphersuite, which determines the above options

Notation and primitives used:
Elliptic curve operations are written in additive notation, with P+Q denoting point addition and x*P denoting scalar multiplication of a point P by a scalar x
x^y - a raised to the power b
x*y - a multiplied by b
|| - octet string concatenation
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 integer. Specified in .

Type conversions:
int_to_string(a, len) - conversion of nonnegative integer a to
to octet string of length len as specified in .
string_to_int(a_string) - conversion of an octet string a_string
to a nonnegative integer as specified in .
point_to_string - conversion of EC point to an ptLen-octet string
as specified in
string_to_point - conversion of an ptLen-octet string to EC point
as specified in .
string_to_point returns INVALID if the octet string does not convert to a valid EC point.
arbitrary_string_to_point - conversion of an arbitrary octet string to an
EC point as specified in
Note that with certain software libraries
(for big integer and elliptic curve arithmetic),
the int_to_string and point_to_string conversions are not needed.
For example, in some implementations, EC point
operations will take octet strings as inputs and
produce octet strings as outputs, without introducing
a separate elliptic curve point type.

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 = x*B - VRF public key, an EC point

ECVRF_prove(SK, alpha_string)
Input:
SK - VRF private key
alpha_string = input alpha, an octet string

Output:
pi_string - VRF proof, octet string of length ptLen+n+qLen

Steps:
Use SK to derive the VRF secret scalar x and the VRF public key Y = x*B
(this derivation depends on the ciphersuite, as per ;
these values can be cached, for example, after key generation, and need not be rederived each time)
H = ECVRF_hash_to_curve(suite_string, Y, alpha_string)
h_string = point_to_string(H)
Gamma = x*H
k = ECVRF_nonce_generation(SK, h_string)
c = ECVRF_hash_points(H, Gamma, k*B, k*H)
s = (k + c*x) mod q
pi_string = point_to_string(Gamma) || int_to_string(c, n) || int_to_string(s, qLen)
Output pi_string

ECVRF_proof_to_hash(pi_string)
Input:
pi_string - VRF proof, octet string of length ptLen+n+qLen

Output:
"INVALID", or
beta_string - VRF hash output, octet string of length hLen

Important note:
ECVRF_proof_to_hash should be run only on pi_string that is known to have been produced by ECVRF_prove, or
from within ECVRF_verify as specified in .

Steps:
D = ECVRF_decode_proof(pi_string)
If D is "INVALID", output "INVALID" and stop
(Gamma, c, s) = D
three_string = 0x03 = int_to_string(3, 1), a single octet with value 3
beta_string = Hash(suite_string || three_string || point_to_string(cofactor * Gamma))
Output beta_string

ECVRF_verify(Y, pi_string, alpha_string)
Input:
Y - public key, an EC point
pi_string - VRF proof, octet string of length ptLen+n+qLen
alpha_string - VRF input, octet string

Output:
("VALID", beta_string), where beta_string is the VRF hash output, octet string of length hLen; or
"INVALID"

Steps:
D = ECVRF_decode_proof(pi_string)
If D is "INVALID", output "INVALID" and stop
(Gamma, c, s) = D
H = ECVRF_hash_to_curve(suite_string, Y, alpha_string)
U = s*B - c*Y
V = s*H - c*Gamma
c' = ECVRF_hash_points(H, Gamma, U, V)
If c and c' are equal, output ("VALID", ECVRF_proof_to_hash(pi_string));
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 the VRF input alpha is used in
for 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_string, Y, alpha_string) 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_string.
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.
However, because the running time of algorithm depends on alpha_string,
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_string, Y, alpha_string)
Input:
suite_string - a single octet specifying ECVRF ciphersuite.
Y - public key, an EC point
alpha_string - value to be hashed, an octet string

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

Steps:
ctr = 0
PK_string = point_to_string(Y)
one_string = 0x01 = int_to_string(1, 1), a single octet with value 1
H = "INVALID"
While H is "INVALID" or H is EC point at infinity:
ctr_string = int_to_string(ctr, 1)
hash_string = Hash(suite_string || one_string || PK_string || alpha_string || ctr_string)
H = arbitrary_string_to_point(hash_string)
If H is not "INVALID" and cofactor > 1, set H = cofactor * H
ctr = ctr + 1

Output H

The following ECVRF_hash_to_curve_elligator2_25519(suite_string, Y, alpha_string)
algorithm implements ECVRF_hash_to_curve using the elligator2
algorithm from Section 5 of (see also ) exclusively for the Ed25519 elliptic curve (which the Edwards equivalent of Curve25519). It can be implemented with running time that is independent of the input alpha (so-called "constant-time").
ECVRF_hash_to_curve_elligator2_25519(suite_string, Y, alpha_string)
Input:
suite_string - a single octet specifying ECVRF ciphersuite.
alpha_string - value to be hashed, an octet string
Y - public key, an EC point

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

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

Constraints on options:
output length of Hash is at least 16n (i.e., 256) bits

Steps:
PK_string = point_to_string(Y)
one_string = 0x01 = int_to_string(1, 1)
(a single octet with value 1)
hash_string = Hash(suite_string || one_string || PK_string || alpha_string )
truncated_h_string = hash_string[0]...hash_string[31]
oneTwentySeven_string = 0x7F = int_to_string(127, 1)
(a single octet with value 127)
truncated_h_string[31] = truncated_h_string[31] & oneTwentySeven_string
(this step clears the high-order bit of octet 31)
r = string_to_int(truncated_h_string)
u = - A / (1 + 2*(r^2) ) mod p
(note: the inverse of (1+2*(r^2)) modulo p is guaranteed to exist)
w = u * (u^2 + A*u + 1) mod p
(this step evaluates the Montgomery equation for Curve25519)
Let e equal the Legendre symbol of w and p
(see note below on how to compute e)
If e is equal to 1 then final_u = u; else final_u = (-A - u) mod p
(note: final_u is the Montgomery u-coordinate of the output; see note below on how to compute it)
y_coordinate = (final_u - 1) / (final_u + 1) mod p
(note 1: y_coordinate is the Edwards coordinate corresponding to final_u)
(note 2: the inverse of (final_u + 1) modulo p is guaranteed to exist)
h_string = int_to_string (y_coordinate, 32)
H_prelim = string_to_point(h_string)
(note: string_to_point will not return INVALID by correctness of Elligator2)
Set H = cofactor * H_prelim
Output H

In order to make this algorithm run in time that is (almost) independent
of the input alpha_string (so-called "constant-time"), implementers should pay particular attention
to Steps 10 and 11 above.
These steps can be implemented using the following approach:
e = w ^ ((p-1)/2) mod p
final_u = (e*u + (e-1) * (A/2)) mod p

The first step will produce a value e that is either 1 or p-1 (it is guaranteed not to be any other value, because w is guaranteed to be nonzero). Implementers should also ensure that the second step
runs in the same amount of time regardless of e by ensuring that
arithmetic in constant time.
Alternatively, let CMOV(result_if_1, result_if_0, selector) be the function that returns result_if_1 when
selector is 1 and result_if_0 when selector is 0. If CMOV is implemented in constant time, then steps 12 and 13 above can be implemented as follows:
e = (w^((p-1)/2))+1 mod p
b = e/2
other_u = (-A-u) mod p
final_u = CMOV(u, other_u, b)

(Note that after the first step, e is either 0 or 2, and only the least significant byte of e is needed in the second step).
CMOV can be implemented in constant time a variety of ways; for example, by expanding b from a single bit to an all-0 or all-1 string (accomplished by negating b in standard two's-complement arithmetic) and then applying bitwise XOR and AND operations as follows: other_x XOR ((x XOR other_x) AND b)
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 .
The following ECVRF_hash_to_curve_Simplified_SWU(suite_string, Y, alpha_string)
algorithm implements ECVRF_hash_to_curve using the simplified Shallue-Woestijne and Ulas
algorithm from Section 7 of (see also ). It can be implemented with running time that is independent of the input alpha (so-called "constant-time"). Generally,
this method can be used
for any curve with prime p that is congruent
to 3 modulo 4; however, the (very unlikely) case of d=0 in step 6 below may need to be handled differently depending on the curve equation, to ensure that the result is a point on the curve.
ECVRF_hash_to_curve_Simplified_SWU(suite_string, Y, alpha_string)
Input:
suite_string - a single octet specifying ECVRF ciphersuite.
alpha_string - value to be hashed, an octet string
Y - public key, an EC point

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

Fixed options:
a and b, constants for the Weierstrass form elliptic curve equation y^2 = x^3 + ax +b for the curve E

Steps:
PK_string = EC2OSP(Y)
one_string = 0x01 = I2OSP(1, 1), a single octet with value 1
h_string = Hash(suite_string || one_string || PK_string || alpha_string)
t = string_to_int(h_string) mod p
r = -(t^2) mod p
d = (r^2 + r) mod p
(d is t^4-t^2 mod p)
If d = 0 then d_inverse = 0; else d_inverse = 1/d mod p
(as long as Hash is secure, the case of d = 0 is an utterly improbably occurrence;
the two cases can be combined into one by computing d_inverse = d^(p-2) mod p)
x = ((-b/a) * (1 + d_inverse)) mod p
w = (x^3 + a*x + b) mod p
(this step evaluates the curve equation)
Let e equal the Legendre symbol of w and p
(see note below on how to compute e)
If e is equal to 0 or 1 then final_x = x; else final_x = r * x mod p
(final_x is the x-coordinate of the output; see note below on how to compute it)
H_prelim = arbitrary_string_to_point(int_to_string(final_x, 2n))
(note: arbitrary_string_to_point will not return INVALID by correctness of Simple SWU)
If cofactor > 1, set H = cofactor * H; else set H = H_prelim
Output H

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 10 and 11 above.
These steps can be implemented using the following approach.
Let CMOV(result_if_1, result_if_0, selector) be the function that returns result_if_1 when
selector is 1 and result_if_0 when selector is 0. If arithmetic and CMOV are implemented in constant time, then steps 9 and 10 above can be implemented as follows:
e = (w ^ ((p-1)/2))+1 mod p
(At this point, e is 0, 1, or 2, as an integer.)
Let b = (e+1) / 2, where / denotes integer division with rounding down.
(Note carefully that this step is integer, not modular, division. Only the last byte of e is needed for this step. This step converts 0, 1, or 2 to 0 or 1.)
other_x = r * x mod p
final_x = CMOV(x, other_x, b)

CMOV can be implemented in constant time a variety of ways; for example, by expanding b from a single bit to an all-0 or all-1 string (accomplished by negating b in standard two's-complement arithmetic) and then applying bitwise XOR and AND operations as follows: other_x XOR ((x XOR other_x) AND b)
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 .
The following subroutines generate the
nonce value k in a deterministic pseudorandom fashion.
ECVRF_nonce_generation_RFC6979(SK, h_string)
Input:
SK - an ECVRF secret key
h_string - an octet string

Output:
k - an integer between 1 and q-1

The ECVRF_nonce_generation function is as specified in
Section 3.2 where
Step a is omitted
h_1 is set equal to h_string
The "suitable for DSA or ECDSA" check in step h.3 is omitted
The hash function H is Hash and its output length hlen is set as hLen*8
The secret key x is set equal to the VRF secret scalar x
The prime q is the same as in this specification
qlen is the binary length of q, i.e., the smallest integer such that 2^qlen > q
All the other values and primitives as defined in

The following is from Steps 2-3 of Section 5.1.6
in .
ECVRF_nonce_generation_RFC8032(SK, h_string)
Input:
SK - an ECVRF secret key
h_string - an octet string

Output:
k - an integer between 0 and q-1

Steps:
hashed_sk_string = Hash (SK)
truncated_hashed_sk_string = hashed_sk_string[32]...hashed_sk_string[63]
k_string = Hash(truncated_hashed_sk_string || h_string)
k = string_to_int(k_string) mod q

ECVRF_hash_points(P1, P2, ..., PM)
Input:
P1...PM - EC points in G

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

Steps:
two_string = 0x02 = int_to_string(2, 1), a single octet with value 2
Initialize str = suite_string || two_string
for PJ in [P1, P2, ... PM]:
str = str || point_to_string(PJ)
c_string = Hash(str)
truncated_c_string = c_string[0]...c_string[n-1]
c = string_to_int(truncated_c_string)
Output c

ECVRF_decode_proof(pi_string)
Input:
pi_string - VRF proof, octet string (ptLen+n+qLen octets)

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

Steps:
let gamma_string = pi_string[0]...p_string[ptLen-1]
let c_string = pi_string[ptLen]...pi_string[ptLen+n-1]
let s_string =pi_string[ptLen+n]...pi_string[ptLen+n+qLen-1]
Gamma = string_to_point(gamma_string)
if Gamma = "INVALID" output "INVALID" and stop.
c = string_to_int(c_string)
s = string_to_int(s_string)
Output Gamma, c, and s

This document defines ECVRF-P256-SHA256-TAI as follows:
suite_string = 0x01 = int_to_string(1, 1).
The EC group G is the NIST P-256 elliptic curve, with curve parameters
as specified in (Section D.1.2.3)
and (Section 2.6). For this group,
2n = qLen = 32 and cofactor = 1.
The key pair generation primitive is specified in
Section 3.2.1 of (q, B, 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 .
The int_to_string function is the I2OSP function specified in Section
4.1 of . (This is big endian representation.)
The string_to_int function is the OS2IP function specified in Section
4.2 of . (This is big endian representation.)
The point_to_string function converts an EC point to an octet string
according to the encoding specified in Section 2.3.3 of
with point compression on.
This implies ptLen = 2n + 1 = 33.
(Note that certain software implementations do not introduce a
separate elliptic curve point type and instead directly treat the
EC point as an octet string per above encoding. When using such
an implementation, the point_to_string function
can be treated as the identity function.)
The string_to_point function converts an octet string to an
EC point according to the encoding specified in Section 2.3.4 of
. This function MUST output INVALID if
the octet string does not decode to an EC point.
arbitrary_string_to_point(h_string) = string_to_point(0x02 || h_string)
(where 0x02 is a single
octet with value 2, 0x02=int_to_string(2, 1)). The input h_string 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 , with hLen = 32.
The ECVRF_hash_to_curve function is as specified in .

This document defines ECVRF-P256-SHA256-SWU as follows:
This ciphersuite is identical to ECVRF-P256-SHA256-TAI except that
the ECVRF_hash_to_curve function is as specified in and
suite_string = 0x02 = int_to_string(2, 1).

This document defines ECVRF-ED25519-SHA512-TAI as follows:
suite_string = 0x03 = int_to_string(3, 1).
The EC group G is the Ed25519
elliptic curve with parameters defined in Table 1 of
.
For this group, 2n = qLen = 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 .
The int_to_string function as specified in the first paragraph of
Section 5.1.2 of . (This is little endian representation.)
The string_to_int function interprets the string as an integer in little-endian
representation.
The point_to_string function converts an EC point to an
octect string according to the encoding specified
in Section 5.1.2 of .
This implies ptLen = 2n = 32.
(Note that certain software implementations do not introduce a
separate elliptic curve point type and instead directly treat the
EC point as an octet string per above encoding. When using such
and implementation, the point_to_string
function can be treated as the identity function.)
The string_to_point function converts an octet string to an EC point
according to the encoding specified in Section 5.1.3
of . This function MUST output INVALID if
the octet string does not decode to an EC point.
arbitrary_string_to_point(h_string) = string_to_point(h_string[0]...h_string[31])
The hash function Hash is SHA-512 as specified in , with hLen = 64.
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-TAI except that
the ECVRF_hash_to_curve function is as specified in and
suite_string = 0x04 = int_to_string(4, 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 B, the generator of group G, is untrusted. If the prover is untrusted,
the Verifier MUST
obtain E and B 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_string)
Input:
PK_string - public key, an octet string

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

Steps:
Y = string_to_point(PK_string)
If Y is "INVALID", output "INVALID" and stop
If cofactor*Y is the EC point at infinty, output "INVALID" and stop
Output Y

Note that if the cofactor = 1, then Step 3 need not multiply Y by the cofactor; instead, it suffices
to output "INVALID" if Y is the point at infinity. Moreover, when cofactor>1, it is not necessary to verify
that Y is in the subgroup G; Step 3 suffices. Therefore, if the cofactor is small, the total number
of points that could cause Step 3 to output "INVALID" may be small, and it may be more efficient to simply
check Y against a fixed list of such points. For example, the following algorithm can be used for the Ed25519 curve:
Y = string_to_point(PK_string)
If Y is "INVALID", output "INVALID" and stop
y_string = PK_string
oneTwentySeven_string = 0x7F = int_to_string(127, 1)
(a single octet with value 127)
y_string[31] = y_string[31] & oneTwentySeven_string
(this step clears the high-order bit of octet 31)
bad_pk[0] = int_to_string(0, 32)
bad_pk[1] = int_to_string(1, 32)
bad_y2 = 2707385501144840649318225287225658788936804267575313519463743609750303402022
bad_pk[2] = int_to_string(bad_y2, 32)
bad_pk[3] = int_to_string(p-bad_y2, 32)
bad_pk[4] = int_to_string(p-1, 32)
bad_pk[5] = int_to_string(p, 32)
bad_pk[6] = int_to_string(p+1, 32)
If y_string is in bad_pk[0]...bad_pk[6], output "INVALID" and stop
Output Y

(bad_pk[0], bad_pk[2], bad_pk[3] each match two bad public keys, depending on the sign of the x-coordinate, which was cleared in step 5, in order to make sure that it does not affect the comparison. bad_pk[1] and bad_pk[4] each match one bad public key, because x-coordinate is 0 for these two public keys. bad_pk[5] and bad_pk[6] are simply bad_pk[0] and bad_pk[1] shifted by p, in case the y-coordinate had not been modular reduced by p. There is no need to shift the other bad_pk values by p, because they will exceed 2^255. These bad keys, which represent all points of order 1, 2, 4, and 8, have been obtained by converting the points specified in to Edwards coordinates.)
A reference implementation of ECVRF-P256-SHA256-TAI, ECVRF-P256-SHA256-SWU, ECVRF-ED25519-SHA512-TAI, ECVRF-ED25519-SHA512-Elligator2
is available at . This implementation is neither secure nor especially effecient, but can be used to generate
test vectors.
An implementation of ECVRF-ED25519-SHA512-Elligator2 is available at .
An implementation of an earlier, slightly different, version of RSA-FDH-VRF (SHA-256) and ECVRF-P256-SHA256-TAI was
first developed
as a part of the NSEC5 project and is available
at .
The Key Transparency project at Google
uses a VRF implemention that is similar to
the ECVRF-P256-SHA256-TAI, with a few minor changes
including the use of SHA-512 instead of SHA-256. Its implementation
is available at
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.
Side channel attacks on cryptographic primatives are an important issue.
Here we discuss only one such side channel: timing attacks that can
be used to leak information about the VRF input alpha. Implementers should
take care to avoid side-channel attacks that leak information about
the VRF private key SK (and the nonce k used in the ECVRF).
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-P256-SHA256-SWU and ECVRF-ED25519-SHA512-Elligator2 can be made to
run in time constant in alpha.
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_string octet and the public key
are hashed together with alpha in ECVRF_hash_to_curve, which ensures
that the curve (including the generator B) and the public
key are included indirectly into subsequent hashes.
Hashing is used for different purposes in the two VRFs (namely, in the RSA-FDH-VRF, in MGF1 and in proof_to_hash; in the ECVRF, in hash_to_curve, nonce_generation, hash_points, and proof_to_hash). The
theoretical analysis assumes each of these 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 RSA-FDH-VRF defined in this document, because the first octets
of the input to the hash function used in MGF1 and in proof_to_hash are different.
This is also 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, proof_to_hash, 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, proof_to_hash, and
hash_points are all different

For the RSA VRF, if future designs need to specify variants of the design in this document, such variants should use different first octets in inputs to MGF1 and to the hash funciton used in proof_to_hash, in order to avoid the possibility
that an adversary can obtain a VRF output under one variant, and then claim it was obtained under
another variant
For the elliptic curve VRF, 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_string 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_string 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 B and PK=Y from ECVRF_hash_points because they are already present
via H, which is computed via hash_to_curve using the suite_string (which identifies B) and Y.
03 - Changed Ed25519 conversions to little-endian, to match RFC 8032; added simple key validation for Ed25519; added Simple SWU cipher suite; clarified Elligator and removed the extra x0 bit, to make Montgomery and Edwards Elligator the same; added domain separation for RSA VRF; improved notation throughout; added nonce generation as a section; changed counter in try-and-increment from four bytes to one, to avoid endian issues; renamed try-and-increment ciphersuites to -TAI; added qLen as a separate paremeter; changed output length to hLen for ECVRF, to match RSAVRF; made Verify return beta so unverified proofs don't end
up in proof_to_hash; added test vectors.
04 - Clarified handling of optional arguments x and PK in ECVRF_prove. Edited implementation status to bring it up to date.

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, David C. Lawerence, Trevor Perrin, Annie Yousar, Stanislav Smyshlyaev, Liliya Akhmetzyanova,
Tony Arcieri, Sergey Gorbunov, Sam Scott, Nick Sullivan, Christopher Wood, Marek Jankowski, Derek Ting-Haye Leung, Adam Suhl, and Gary Belvin 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
Elligator: elliptic-curve points indistinguishable from uniform random strings
Efficient Indifferentiable Hashing into Ordinary Elliptic Curves
Construction of rational points on elliptic curves over finite fields
Rational points on certain hyperelliptic curves over finite fields
How do I validate Curve25519 public keys?
Algorand: Scaling Byzantine Agreements for Cryptocurrencies
Ouroboros: A Provably Secure Proof-of-Stake Blockchain Protocol
Public Key Cryptography for the Financial Services Industry: The Elliptic Curve Digital Signature Algorithm (ECDSA)
The test vectors in this section were genereated using the reference implementation at .
These two example secret keys and messages are taken from Appendix A.2.5 of .
SK = x = c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
PK = 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
alpha = 73616d706c65 (ASCII "sample")
try_and_increment succeded on ctr = 0
H = 02e2e1ab1b9f5a8a68fa4aad597e7493095648d3473b213bba120fe42d1a595f3e
k = c1aba586552242e6b324ab4b7b26f86239226f3cfa85b1c3b675cc061cf147dc
U = k*B = 02007fe22a3ed063db835a63a92cb1e487c4fea264c3f3700ae105f8f3d3fd391f
V = k*H = 03d0a63fa7a7fefcc590cb997b21bbd21dc01304102df183fb7115adf6bcbc2a74
pi = 029bdca4cc39e57d97e2f42f88bcf0ecb1120fb67eb408a856050dbfbcbf57c524193b7a850195ef3d5329018a8683114cb446c33fe16ebcc0bc775b043b5860dcb2e553d91268281688438df9394103ab
beta = 59ca3801ad3e981a88e36880a3aee1df38a0472d5be52d6e39663ea0314e594c
SK = x = c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
PK = 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
alpha = 74657374 (ASCII "test")
try_and_increment succeded on ctr = 0
H = 02ca565721155f9fd596f1c529c7af15dad671ab30c76713889e3d45b767ff6433
k = 7fc43fbc2aa51886139614792c613e672624b3fb8d0cf3fa6f52543d6a2fc26c
U = k*B = 037cd79cd8ab7b0324600b4d76c673a0726f3f1ff26b4b850d0c14b3aa272bf841
V = k*H = 02372cc8b56281e8d7f21ab7503c10af22164a4227e7433de0953a0df5e7a609bd
pi = 03873a1cce2ca197e466cc116bca7b1156fff599be67ea40b17256c4f34ba2549c9c8b100049e76661dbcf6393e4d625597ed21d4de684e08dc6817b60938f3ff4148823ea46a47fa8a4d43f5fa6f77dc8
beta = dc85c20f95100626eddc90173ab58d5e4f837bb047fb2f72e9a408feae5bc6c1
This example secret key and message are taken from Appendix L.4.2 of .
SK = x = 2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8
PK = 03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d
alpha = 4578616d706c65206f66204543445341207769746820616e736970323536723120616e64205348412d323536 (ASCII "Example of ECDSA with ansip256r1 and SHA-256")
try_and_increment succeded on ctr = 1
H = 02141e41d4d55802b0e3adaba114c81137d95fd3869b6b385d4487b1130126648d
k = 111e1505c8531c885dab6607a0962cd40a0af77637cdf183c7c9fb799dded43e
U = k*B = 02b3aceb619b90e811c8e50de73b27e65dd84669821055cb60dc1fa47199396c74
V = k*H = 02f117fe7daa8942d5492cc968784ced16025161b2dad374808eb7fbaf5eda5331
pi = 02abe3ce3b3aa2ab3c6855a7e729517ebfab6901c2fd228f6fa066f15ebc9b9d41fd212750d9ff775527943049053a77252e9fa59e332a2e5d5db6d0be734076e98befcdefdcbaf817a5c13d4e45fbf9bc
beta = e880bde34ac5263b2ce5c04626870be2cbff1edcdadabd7d4cb7cbc696467168
These two example secret keys and messages are taken from Appendix A.2.5 of .
SK = x = c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
PK = 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
alpha = 73616d706c65 (ASCII "sample")
In SWU: t = f1523667d029b9119a319a5bb316ff846691600e3552514ec4f93f9c84d65a4f
In SWU: w = d8125c3ae82fc2b7f1c326b6f3dbfdf3583272336a60cb08efb84e002e98a3b3
In SWU: e = -1
H = 027827143876a58c2189402306c6ff6f7f9a7271067f3ed28eb63790d58a84fdd6
k = e15d8e7677f9473ae922d36977dbcc305a4ffd8149499dccfcb44fa097a2200c
U = k*B = 035dc0dee6903dba1a3e7cae4e2a960609e873e6e696cc8d5e56dfa8efccdfc97a
V = k*H = 03bec301c2930d69ed359eab2a54349d1431625c7a3ee0cfc2643ae5f8a21c3add
pi = 021d684d682e61dd76c794eef43988a2c61fbdb2af64fbb4f435cc2a842b0024c35641fe838a72d0d9bc1bcf032f895f3b3f4c79d0f8f9d5705d83181fe82e19f49619eb8290930809b2b9651786e4f945
beta = 143f36bf7175053315693cfcfdff5aebb13e5eb9c47f897f53f81561993cfcd2
SK = x = c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
PK = 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
alpha = 74657374 (ASCII "test")
In SWU: t = e20da1d7386cb673deffec63d47ec65862dce55f113be168fa45cba2a6c1ddbc
In SWU: w = 0eed10be2937c902c9612d80b8ea5b0783f81c419faedd57efc84e6dfcfe2c72
In SWU: e = 1
H = 020e6c14efc8bc7150a3467aafa78be9856a2c6e405bdcc50f767fe638569d0172
k = 2addf2924eea6557e87acd635f08b54156cba70d718d8a3b6268af795cfdb7f2
U = k*B = 028347e823490cb50becb229cf059942afff39d6b276b987a384e45f29d1bf0dc3
V = k*H = 034759927f83caf0ed5d7ad1505844548051ef90a2f29de30efaf8eb811afb3342
pi = 0376b758f457d2cabdfaeb18700e46e64f073eb98c119dee4db6c5bb1eaf677806895ab451335f6adb792d40c68351929fce44068ffdcbbeac12f058b0365856ed5d86aadba1f54c9db13f9c8759589609
beta = 6b5bb622a6bc1387a7dcc4f46cfdcc3bce67669b32f3bc39e047c3b6cd3e65d9
This example secret key and message are taken from Appendix L.4.2 of .
SK = x = 2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8
PK = 03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d
alpha = 4578616d706c65206f66204543445341207769746820616e736970323536723120616e64205348412d323536 (ASCII "Example of ECDSA with ansip256r1 and SHA-256")
In SWU: t = e93da6ba2bca714061dc94c8c513343ad11bfc9678339e4a8bd86a08232aa6d7
In SWU: w = 76f564cca31934c80dd2a285ba43543df63a078b132c8f34d2ab1b7089cb3401
In SWU: e = -1
H = 02429690b91e1783cd0d7e393db07cc44b48c226cb837adb2282251cabf431a484
k = 2ba4cd5e9f7be946659555b3d052af618d2d3a14f5a756c8d56fab058d1831ff
U = k*B = 025943c217f48354e297156ac7dce8d2b50f63867acc23ba1aea87a66578392ca8
V = k*H = 0399ccafce764acf0db264183d97c1ae968daf661b9931a145bb8cbfc5f3f8f9a5
pi = 035e844533a7c5109ab3dffd04f2ef0d38d679101124f15243199ce92f0f29477cd29f8754f3bbdea3dd129560e9ba0c73ae7894a8d0c0e1ac01e5c2685da67009d96e6ccdb634c7e0c5f38fa3e4908c02
beta = be1dcb17e9815ac6acf819e7ad4b75e575eafad25915c2608959d780364fc912
These three example secret keys and messages are taken from Section 7.1 of .
SK = 9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60
PK = d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a
alpha = (the empty string)
x = 307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f
try_and_increment succeded on ctr = 0
H = 5b2c80db3ce2d79cc85b1bfb269f02f915c5f0e222036dc82123f640205d0d24
k = 647ac2b3ca3f6a77e4c4f4f79c6c4c8ce1f421a9baaa294b0adf0244915130f7067640acb6fd9e7e84f8bc30d4e03a95e410b82f96a5ada97080e0f187758d38
U = k*B = a21c342b8704853ad10928e3db3e58ede289c798e3cdfd485fbbb8c1b620604f
V = k*H = 426fe41752f0b27439eb3d0c342cb645174a720cae2d4e9bb37de034eefe27ad
pi = 9275df67a68c8745c0ff97b48201ee6db447f7c93b23ae24cdc2400f52fdb08a1a6ac7ec71bf9c9c76e96ee4675ebff60625af28718501047bfd87b810c2d2139b73c23bd69de66360953a642c2a330a
beta = a64c292ec45f6b252828aff9a02a0fe88d2fcc7f5fc61bb328f03f4c6c0657a9d26efb23b87647ff54f71cd51a6fa4c4e31661d8f72b41ff00ac4d2eec2ea7b3
SK = 4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb
PK = 3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c
alpha = 72 (1 byte)
x = 68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51
try_and_increment succeded on ctr = 4
H = 08e18a34f3923db32e80834fb8ced4e878037cd0459c63ddd66e5004258cf76c
k = 627237308294a8b344a09ad893997c630153ee514cd292eddd577a9068e2a6f24cbee0038beb0b1ee5df8be08215e9fc74608e6f9358b0e8d6383b1742a70628
U = k*B = 18b5e500cb34690ced061a0d6995e2722623c105221eb91b08d90bf0491cf979
V = k*H = 87e1f47346c86dbbd2c03eafc7271caa1f5307000a36d1f71e26400955f1f627
pi = 84a63e74eca8fdd64e9972dcda1c6f33d03ce3cd4d333fd6cc789db12b5a7b9d03f1cb6b2bf7cd81a2a20bacf6e1c04e59f2fa16d9119c73a45a97194b504fb9a5c8cf37f6da85e03368d6882e511008
beta = cddaa399bb9c56d3be15792e43a6742fb72b1d248a7f24fd5cc585b232c26c934711393b4d97284b2bcca588775b72dc0b0f4b5a195bc41f8d2b80b6981c784e
SK = c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7
PK = fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025
alpha = af82 (2 bytes)
x = 909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c
try_and_increment succeded on ctr = 0
H = e4581824b70badf0e57af789dd8cf85513d4b9814566de0e3f738439becfba33
k = a950f736af2e3ae2dbcb76795f9cbd57c671eee64ab17069f945509cd6c4a74852fe1bbc331e1bd573038ec703ca28601d861ad1e9684ec89d57bc22986acb0e
U = k*B = 5114dc4e741b7c4a28844bc585350240a51348a05f337b5fd75046d2c2423f7a
V = k*H = a6d5780c472dea1ace78795208aaa05473e501ed4f53da57e1fb13b7e80d7f59
pi = aca8ade9b7f03e2b149637629f95654c94fc9053c225ec21e5838f193af2b727b84ad849b0039ad38b41513fe5a66cdd2367737a84b488d62486bd2fb110b4801a46bfca770af98e059158ac563b690f
beta = d938b2012f2551b0e13a49568612effcbdca2aed5d1d3a13f47e180e01218916e049837bd246f66d5058e56d3413dbbbad964f5e9f160a81c9a1355dcd99b453
These three example secret keys and messages are taken from Section 7.1 of .
SK = 9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60
PK = d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a
alpha = (the empty string)
x = 307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f
In Elligator: r = 9ddd071cd5837e591a3a40c57a46701bb7f49b1b53c670d490c2766a08fa6e3d
In Elligator: w = c7b5d6239e52a473a2b57a92825e0e5de4656e349bb198de5afd6a76e5a07066
In Elligator: e = -1
H = 1c5672d919cc0a800970cd7e05cb36ed27ed354c33519948e5a9eaf89aee12b7
k = 868b56b8b3faf5fc7e276ff0a65aaa896aa927294d768d0966277d94599b7afe4a6330770da5fdc2875121e0cbecbffbd4ea5e491eb35be53fa7511d9f5a61f2
U = k*B = c4743a22340131a2323174bfc397a6585cbe0cc521bfad09f34b11dd4bcf5936
V = k*H = e309cf5272f0af2f54d9dc4a6bad6998a9d097264e17ae6fce2b25dcbdd10e8b
pi = b6b4699f87d56126c9117a7da55bd0085246f4c56dbc95d20172612e9d38e8d7ca65e573a126ed88d4e30a46f80a666854d675cf3ba81de0de043c3774f061560f55edc256a787afe701677c0f602900
beta = 5b49b554d05c0cd5a5325376b3387de59d924fd1e13ded44648ab33c21349a603f25b84ec5ed887995b33da5e3bfcb87cd2f64521c4c62cf825cffabbe5d31cc
SK = 4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb
PK = 3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c
alpha = 72 (1 byte)
x = 68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51
In Elligator: r = 92181bd612695e464049590eb1f9746750d6057441789c9759af8308ac77fd4a
In Elligator: w = 7ff6d8b773bfbae57b2ab9d49f9d3cb7d9af40a03d3ed3c6beaaf2d486b1fe6e
In Elligator: e = 1
H = 86725262c971bf064168bca2a87f593d425a49835bd52beb9f52ea59352d80fa
k = fd919e9d43c61203c4cd948cdaea0ad4488060db105d25b8fb4a5da2bd40e4b8330ca44a0538cc275ac7d568686660ccfd6323c805b917e91e28a4ab352b9575
U = k*B = 04b1ba4d8129f0d4cec522b0fd0dff84283401df791dcc9b93a219c51cf27324
V = k*H = ca8a97ce1947d2a0aaa280f03153388fa7aa754eedfca2b4a7ad405707599ba5
pi = ae5b66bdf04b4c010bfe32b2fc126ead2107b697634f6f7337b9bff8785ee111200095ece87dde4dbe87343f6df3b107d91798c8a7eb1245d3bb9c5aafb093358c13e6ae1111a55717e895fd15f99f07
beta = 94f4487e1b2fec954309ef1289ecb2e15043a2461ecc7b2ae7d4470607ef82eb1cfa97d84991fe4a7bfdfd715606bc27e2967a6c557cfb5875879b671740b7d8
SK = c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7
PK = fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025
alpha = af82 (2 bytes)
x = 909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c
In Elligator: r = dcd7cda88d6798599e07216de5a48a27dcd1cde197ab39ccaf6a906ae6b25c7f
In Elligator: w = 2ceaa2c2ff3028c34f9fbe076ff99520b925f18d652285b4daad5ccc467e523b
In Elligator: e = -1
H = 9d8663faeb6ab14a239bfc652648b34f783c2e99f758c0e1b6f4f863f9419b56
k = 8f675784cdc984effc459e1054f8d386050ec400dc09d08d2372c6fe0850eaaa50defd02d965b79930dcbca5ba9222a3d99510411894e63f66bbd5d13d25db4b
U = k*B = d6f8a95a4ce86812e3e50febd9d48196b3bc5d1d9fa7b6dfa33072641b45d029
V = k*H = f77cd4ce0b49b386e80c3ce404185f93bb07463600dc14c31b0a09beaff4d592
pi = dfa2cba34b611cc8c833a6ea83b8eb1bb5e2ef2dd1b0c481bc42ff36ae7847f6ab52b976cfd5def172fa412defde270c8b8bdfbaae1c7ece17d9833b1bcf31064fff78ef493f820055b561ece45e1009
beta = 2031837f582cd17a9af9e0c7ef5a6540e3453ed894b62c293686ca3c1e319dde9d0aa489a4b59a9594fc2328bc3deff3c8a0929a369a72b1180a596e016b5ded