Network Working Group S. Goldberg
Internet-Draft Boston University
Intended status: Standards Track D. Papadopoulos
Expires: September 14, 2017 University of Maryland
J. Vcelak
NS1
March 13, 2017
Verifiable Random Functions (VRFs)
draft-goldbe-vrf-00
Abstract
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).
Status of This Memo
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Rationale . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2. Requirements . . . . . . . . . . . . . . . . . . . . . . 3
1.3. Terminology . . . . . . . . . . . . . . . . . . . . . . . 3
2. VRF Algorithms . . . . . . . . . . . . . . . . . . . . . . . 3
3. VRF Security Properties . . . . . . . . . . . . . . . . . . . 4
3.1. Full Uniqueness or Trusted Uniqueness . . . . . . . . . . 4
3.2. Full Pseudorandomness or Selective Pseudorandomness . . . 5
3.3. Full Collison Resistance or Trusted Collision Resistance 5
4. RSA Full Domain Hash VRF (RSA-FDH-VRF) . . . . . . . . . . . 6
4.1. RSA-FDH-VRF Proving . . . . . . . . . . . . . . . . . . . 7
4.2. RSA-FDH-VRF Proof To Hash . . . . . . . . . . . . . . . . 7
4.3. RSA-FDH-VRF Verifying . . . . . . . . . . . . . . . . . . 8
5. Elliptic Curve VRF (EC-VRF) . . . . . . . . . . . . . . . . . 8
5.1. EC-VRF Proving . . . . . . . . . . . . . . . . . . . . . 10
5.2. EC-VRF Proof To Hash . . . . . . . . . . . . . . . . . . 10
5.3. EC-VRF Verifying . . . . . . . . . . . . . . . . . . . . 11
5.4. EC-VRF Auxiliary Functions . . . . . . . . . . . . . . . 11
5.4.1. EC-VRF Hash To Curve . . . . . . . . . . . . . . . . 11
5.4.2. EC-VRF Hash Points . . . . . . . . . . . . . . . . . 13
5.4.3. EC-VRF Decode Proof . . . . . . . . . . . . . . . . . 13
5.5. EC-VRF Ciphersuites . . . . . . . . . . . . . . . . . . . 14
6. Implementation Status . . . . . . . . . . . . . . . . . . . . 15
7. Security Considerations . . . . . . . . . . . . . . . . . . . 15
7.1. Key Generation . . . . . . . . . . . . . . . . . . . . . 15
7.2. Proper randomness for EC-VRF . . . . . . . . . . . . . . 16
7.3. Timing attacks . . . . . . . . . . . . . . . . . . . . . 16
7.4. Selective vs Full Pseudorandomness . . . . . . . . . . . 16
8. Change Log . . . . . . . . . . . . . . . . . . . . . . . . . 17
9. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 17
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 17
10.1. Normative References . . . . . . . . . . . . . . . . . . 17
10.2. Informative References . . . . . . . . . . . . . . . . . 18
Appendix A. Open Issues . . . . . . . . . . . . . . . . . . . . 19
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 19
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1. Introduction
1.1. Rationale
A Verifiable Random Function (VRF) [MRV99] 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.
1.2. Requirements
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 [RFC2119].
1.3. Terminology
The following terminology is used through this document:
SK: The private key for the VRF.
PK: The public key for the VRF.
alpha: The input to be hashed by the VRF.
beta: The VRF hash output.
pi: The VRF proof.
2. VRF Algorithms
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:
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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.
3. VRF Security Properties
VRFs are designed to ensure the following security properties.
3.1. Full Uniqueness or Trusted Uniqueness
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
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generated in a trustworthy manner. In otherwords, uniqueness might
not hold if keys were generated in an invalid manner.
3.2. Full Pseudorandomness or Selective Pseudorandomness
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.)
3.3. Full Collison Resistance or Trusted Collision Resistance
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
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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.
4. RSA Full Domain Hash VRF (RSA-FDH-VRF)
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 Section 3. Its security follows
from the standard RSA assumption in the random oracle model. Formal
security proofs are in [nsec5ecc].
The VRF computes the proof pi as a deterministic RSA signature on
input alpha using the RSA Full Domain Hash Algorithm [RFC8017]
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 [RFC8017].
In this document, the notation from [RFC8017] 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:
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I2OSP - Coversion of a nonnegative integer to an octet string as
defined in Section 4.1 of [RFC8017]
OS2IP - Coversion of an octet string to a nonnegative integer as
defined in Section 4.2 of [RFC8017]
RSASP1 - RSA signature primitive as defined in Section 5.2.1 of
[RFC8017]
RSAVP1 - RSA verification primitive as defined in Section 5.2.2 of
[RFC8017]
MGF1 - Mask Generation Function based on a hash function as
defined in Section B.2.1 of [RFC8017]
4.1. RSA-FDH-VRF Proving
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:
1. EM = MGF1(alpha, k - 1)
2. m = OS2IP(EM)
3. s = RSASP1(K, m)
4. pi = I2OSP(s, k)
5. Output pi
4.2. RSA-FDH-VRF Proof To Hash
RSAFDHVRF_proof2hash(pi)
Input:
pi - proof, an octet string of length k
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Output:
beta - VRF hash output, an octet string of length hLen
Steps:
1. beta = Hash(pi)
2. Output beta
4.3. RSA-FDH-VRF Verifying
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:
1. s = OS2IP(pi)
2. m = RSAVP1((n, e), s)
3. EM = I2OSP(m, k - 1)
4. EM' = MGF1(alpha, k - 1)
5. If EM and EM' are the same, output "VALID"; else output
"INVALID".
5. Elliptic Curve VRF (EC-VRF)
The Elliptic Curve Verifiable Random Function (EC-VRF) is VRF that
satisfies the trusted uniqueness, full pseudorandomness, and trusted
collision resistance properties defined in Section 3. 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 [nsec5ecc].
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The key pair generation primitive is specified in Section 3.2.1 of
[SECG1].
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 [SECG1]
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OS2ECP - octet string to EC point conversion with point
compression as specified in Section 2.3.4 of [SECG1]
5.1. EC-VRF Proving
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:
1. h = ECVRF_hash_to_curve(alpha, g^x)
2. gamma = h^x
3. choose a random nonce k from [0, q-1]
4. c = ECVRF_hash_points(g, h, g^x, h^x, g^k, h^k)
5. s = k - c*q mod q
6. pi = ECP2OS(gamma) || I2OSP(c, n) || I2OSP(s, 2n)
7. Output pi
5.2. EC-VRF Proof To Hash
ECVRF_proof2hash(pi)
Input:
pi - VRF proof, octet string of length 5n+1
Output:
beta - VRF hash output, octet string of length 2n
Steps:
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1. beta = pi[2] || pi[3] || ... pi[2n+1]
2. Output beta
5.3. EC-VRF Verifying
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:
1. gamma, c, s = ECVRF_decode_proof(pi)
2. If gamma is not a valid EC point in G, output "INVALID" and stop.
3. u = (g^x)^c * g^s
4. h = ECVRF_hash_to_curve(alpha, g^x)
5. v = gamma^c * h^s
6. c' = ECVRF_hash_points(g, h, g^x, gamma, u, v)
7. If c and c' are the same, output "VALID"; else output "INVALID".
5.4. EC-VRF Auxiliary Functions
5.4.1. EC-VRF Hash To Curve
The ECVRF_hash_to_curve algorithm takes in an octet string alpha and
converts it to h, an EC point in G.
5.4.1.1. ECVRF_hash_to_curve1
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.
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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 [Icart09]).
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:
1. ctr = 0
2. pk = ECP2OS(g^x)
3. Repeat:
A. CTR = I2OSP(ctr, 4)
B. p = 0x02 || Hash(alpha || pk || CTR)
C. Goto step 3 if OS2ECP(p) is valid EC point in G
D. p = 0x03 || Hash(alpha || pk || CTR)
E. Goto step 3 if OS2ECP(p) is valid EC point in G
F. ctr = ctr + 1
4. h = OS2ECP(p)
5. 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 [SECG1]. Similarly, the 0x03 octet in step D
represents negative y-coefficient.
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5.4.1.2. ECVRF_hash_to_curve2
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 [Icart09]. ]
5.4.2. EC-VRF Hash Points
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:
1. P = ""
2. for p_i in [p_1, p_2, ... p_j]:
P = P || ECP2OS(p_i)
3. h' = Hash(P)
4. h = OS2IP(h'[1] || h'[2] || ... h'[n])
5. Output h
5.4.3. EC-VRF Decode Proof
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
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s - integer between 0 and 2^(16n)-1
Steps:
1. let gamma', c', s' be pi split after (2n+1)-th and (3n+1)-th
octet
2. gamma = OS2ECP(gamma')
3. c = OS2IP(c')
4. s = OS2IP(s')
5. Output gamma, c, and s
5.5. EC-VRF Ciphersuites
[Seeking feedback on this section!]
This document defines EC-VRF-P256-SHA256 as follows:
o The EC group G is the NIST-P256 elliptic curve, with curve
parameters as specified in [FIPS-186-3] (Section D.1.2.3) and
[RFC5114] (Section 2.6). For this group, the length in octets of
a single coordinate of an EC point is 2n = 32.
o The hash function Hash is SHA-256 as specified in [RFC6234].
o The ECVRF_hash_to_curve function is ECVRF_hash_to_curve1, as
specified in Section 5.4.1.1.
This document defines EC-VRF-ED25519-SHA256 as follows:
o The EC group G is the Ed25519 elliptic curve with parameters
defined in [RFC7748] (Section 4.1). For this group, the length in
octets of a single coordinate of an EC point is 2n = 32.
o The hash function Hash is SHA-256 as specified in [RFC6234].
o The ECVRF_hash_to_curve function is as specified in
Section 5.4.1.1.
[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?]
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6. Implementation Status
An implementation of the RSA-FDH-VRF (SHA-256) and EC-VRF-P256-SHA256
was developed as a part of the NSEC5 project [I-D.vcelak-nsec5] 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:
7. Security Considerations
7.1. Key Generation
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 [GQ88] 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.
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7.2. Proper randomness for EC-VRF
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.
7.3. Timing attacks
The EC-VRF_hash_to_curve algorithm defined in Section 5.4.1.1 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.
7.4. Selective vs Full Pseudorandomness
[nsec5ecc] 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.
o 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.
o 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.
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8. Change Log
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.
9. Contributors
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.
10. References
10.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
[RFC8017] Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch,
"PKCS #1: RSA Cryptography Specifications Version 2.2",
RFC 8017, DOI 10.17487/RFC8017, November 2016,
.
[RFC5114] Lepinski, M. and S. Kent, "Additional Diffie-Hellman
Groups for Use with IETF Standards", RFC 5114,
DOI 10.17487/RFC5114, January 2008,
.
[RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms
(SHA and SHA-based HMAC and HKDF)", RFC 6234,
DOI 10.17487/RFC6234, May 2011,
.
[RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
for Security", RFC 7748, DOI 10.17487/RFC7748, January
2016, .
Goldberg, et al. Expires September 14, 2017 [Page 17]
Internet-Draft VRF March 2017
[I-D.vcelak-nsec5]
Vcelak, J., Goldberg, S., Papadopoulos, D., Huque, S., and
D. Lawrence, "NSEC5, DNSSEC Authenticated Denial of
Existence", draft-vcelak-nsec5-04 (work in progress),
March 2017.
[FIPS-186-3]
National Institute for Standards and Technology, "Digital
Signature Standard (DSS)", FIPS PUB 186-3, June 2009.
[SECG1] Standards for Efficient Cryptography Group (SECG), "SEC 1:
Elliptic Curve Cryptography", Version 2.0, May 2009,
.
10.2. Informative References
[nsec5ecc]
Papadopoulos, D., Wessels, D., Huque, S., Vcelak, J.,
Naor, M., Reyzin, L., and S. Goldberg, "NSEC5 from
Elliptic Curves", in ePrint Cryptology Archive 2017/099,
February 2017, .
[MRV99] Michali, S., Rabin, M., and S. Vadhan, "Verifiable Random
Functions", in FOCS, 1999.
[CP92] Chaum, D. and C. Pederson, "Wallet databases with
observers", in FOCS, 1992.
[Icart09] Icart, T., "How to Hash into Elliptic Curves", in CRYPTO,
2009.
[GQ88] Guillou, L. and JJ. Quisquater, "A Practical Zero-
Knowledge Protocol Fitted to Security Microprocessor
Minimizing Both Transmission and Memory", in Advances in
Cryptology - EUROCRYPT '88, 1988.
Goldberg, et al. Expires September 14, 2017 [Page 18]
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Appendix A. Open Issues
Note to RFC Editor: please remove this appendix before publication as
an RFC.
1. Open issues
Authors' Addresses
Sharon Goldberg
Boston University
111 Cummington St, MCS135
Boston, MA 02215
USA
EMail: goldbe@cs.bu.edu
Dimitrios Papadopoulos
University of Maryland
8223 Paint Branch Dr
College Park, MD 20740
USA
EMail: dipapado@bu.edu
Jan Vcelak
NS1
16 Beaver St
New York, NY 10004
USA
EMail: jvcelak@ns1.com
Goldberg, et al. Expires September 14, 2017 [Page 19]