| CFRG | S. Goldberg |
| Internet-Draft | L. Reyzin |
| Intended status: Standards Track | Boston University |
| Expires: December 31, 2018 | D. Papadopoulos |
| Hong Kong University of Science and Techology | |
| J. Vcelak | |
| NS1 | |
| June 29, 2018 |
Verifiable Random Functions (VRFs)
draft-irtf-cfrg-vrf-02
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).
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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.
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 [RFC2119].
The following terminology is used through this document:
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
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
The VRFs defined in this document allow anyone to deterministically obtain the VRF hash output beta directly from the proof value pi as
Notice that this means that
and thus this document will specify VRF_prove and VRF_proof2hash rather than VRF_hash.
The proof pi allows a Verifier holding the public key PK to verify that beta is the correct VRF hash of input alpha under key PK. Thus, the VRF also comes with an algorithm
that outputs VALID if beta=VRF_proof2hash(pi) is the correct VRF hash of alpha under key PK, and outputs INVALID otherwise.
VRFs are designed to ensure the following security properties.
Uniqueness means that, for any fixed public VRF key and for any input alpha, there is a unique VRF output beta that can be proved to be valid. Uniqueness must hold even for an adversarial Prover that knows the VRF private key SK.
More percisely, "full uniqueness" states that a computationally-bounded adversary cannot choose a VRF public key PK, a VRF input alpha, two different VRF hash outputs beta1 and beta2, and two proofs pi1 and pi2 such that VRF_verify(PK, alpha, pi1) and VRF_verify(PK, alpha, pi2) both output VALID.
A slightly weaker security property called "trusted uniquness" sufficies for many applications. Trusted uniqueness is the same as full uniqueness, but it must hold only if the VRF keys PK and SK were generated in a trustworthy manner. In other words, uniqueness might not hold if keys were generated in an invalid manner or with bad randomness.
Like any cryprographic hash function, VRFs need to be collision resistant. Collison resistance must hold even for an adversarial Prover that knows the VRF private key SK.
More percisely, "full collision resistance" states that it should be computationally infeasible for an adversary to find two distinct VRF inputs alpha1 and alpha2 that have the same VRF hash beta, even if that adversary knows the private VRF key SK.
For most applications, a slightly weaker security property called "trusted collision resistance" suffices. Trusted collision resistance is the same as collision resistance, but it holds only if PK and SK were generated in a trustworthy manner.
Pseudorandomness ensures that when an adversarial Verifier sees a VRF hash output beta without its corresponding VRF proof pi, then beta is indistinguishable from a random value.
More percisely, suppose the public and private VRF keys (PK, SK) were generated in a trustworthy manner. Pseudorandomness ensures that the VRF hash output beta (without its corresponding VRF proof pi) on any adversarially-chosen "target" VRF input alpha looks indistinguishable from random for any computationally bounded adversary who does not know the private VRF key SK. This holds even if the adversary also gets to choose other VRF inputs alpha' and observe their corresponding VRF hash outputs beta' and proofs pi'.
With "full pseudorandomness", the adversary is allowed to choose the "target" VRF input alpha at any time, even after it observes VRF outputs beta' and proofs pi' on a variety of chosen inputs alpha'.
"Selective pseudorandomness" is a weaker security property which suffices in many applications. Here, the adversary must choose the target VRF input alpha independently of the public VRF key PK, and before it observes VRF outputs beta' and proofs pi' on inputs alpha' of its choice.
It is important to remember that the VRF output beta does not look random to the Prover, or to any other party that knows the private VRF key SK! Such a party can easily distinguish beta from a random value by comparing beta to the result of VRF_hash(SK, alpha).
Also, the VRF output beta does not look random to any party that knows valid VRF proof pi corresponding to the VRF input alpha, even if this party does not know the private VRF key SK. Such a party can easily distinguish beta from a random value by checking whether VRF_verify(PK, alpha, pi) returns "VALID" and beta = VRF_proof2hash(pi).
Also, the VRF output beta may not look random if VRF key generation was not done in a trustworthy fashion. (For example, if VRF keys were generated with bad randomness.)
[TODO: This property is not needed for applications that use VRFs to prevent enumeration of hash-based data structures. However, we noticed that some other applications of VRF (e.g. Algorand, Oroborus) rely on this property. We are waiting on a formal definition of this property in the literature, and a proof that our ECVRF scheme can satisfy this property. Preliminary analysis suggests that acheiving this property requires ECVRF verifiers to run an VRF_validate_key() key function upon receipt of VRF public keys and the proof2hash function to be modified to take in (gamma, beta, pk) rather than just gamma.]
Pseudorandomness, as defined in Section 3.3, does not hold if the VRF keys were generated adversarially.
There is, however, a different type of pseudorandomness that could hold even if the VRF keys are generated adversarially, as long as the VRF input alpha is unpredictable. This property is similar to the pseudorandomness achieved by an (ordinary, unkeyed) cryptographic hash function.
[TODO: Formal definition here.]
The RSA Full Domain Hash VRF (RSA-FDH-VRF) is a VRF that satisfies the "trusted uniqueness", "trusted collision resistance", and "full pseudorandomness" properties defined in 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.
Parameters used:
Fixed options:
Constraints on options:
Primitives used:
RSAFDHVRF_prove(K, alpha)
Input:
Output:
Steps:
RSAFDHVRF_proof2hash(pi)
Input:
Output:
Steps:
RSAFDHVRF_verify((n, e), alpha, pi)
Input:
Output:
Steps:
The Elliptic Curve Verifiable Random Function (ECVRF) is a VRF that satisfies the trusted uniqueness, trusted collision resistance, and full pseudorandomness properties defined in Section 3. The security of this VRF follows from the decisional Diffie-Hellman (DDH) assumption in the random oracle model. Formal security proofs are in [nsec5ecc].
To additionally satisfy "full uniqueness" and "full collision resistance", the Verifier MUST additionally perform the validation procedure specified in Section 5.6 upon receipt of the public VRF key.
Fixed options (specified in Section 5.5):
Notation and primitives used:
Parameters used (the generation of these parameters is specified in Section 5.5):
Note: this function must have the VRF private key SK as input. Below we make it more efficient by supplying it also with the secret scalar x and the public key y as additional inputs; however, each of these can be computed from SK if desired.
ECVRF_prove(y, x, alpha)
Input:
Output:
Steps:
ECVRF_proof2hash(pi)
Input:
Output:
Steps:
ECVRF_verify(y, pi, alpha)
Input:
Output:
Steps:
The ECVRF_hash_to_curve algorithm takes in the VRF input alpha and converts it to h, an EC point in G. This algorithm is the only place alpha is used in the proving and verfying. See Section 7.6 for further discussion.
The algorithms in this section are not compatible with each other; the choice of algorithm is made in Section 5.5.
The following ECVRF_hash_to_curve_try_and_increment(suite, y, alpha) algorithm implements ECVRF_hash_to_curve in a simple and generic way that works for any elliptic curve.
The running time of this algorithm depends on alpha. For the ciphersuites specified in Section 5.5, this algorithm is expected to find a valid curve point after approximately two attempts (i.e., when ctr=1) on average. See also [Icart09].
However, because the running time of algorithm depends on alpha, this algorithm SHOULD be avoided in applications where it is important that the VRF input alpha remain secret.
ECVRF_hash_to_try_and_increment(suite, y, alpha)
Input:
Output:
Steps:
The following ECVRF__hash_to_curve_elligator2_25519(suite, y, alpha) algorithm implements ECVRF_hash_to_curve using the elligator2 algorithm exclusively for Curve25519.
ECVRF_hash_to_curve_elligator2_25519(y, alpha)
Input:
Output:
Fixed options:
Constraints on options:
Steps:
In order to make this algorithm run in time that is (almost) independent of the input (so-called "constant-time"), implementers should pay particular attention to Steps 7 and 8 above. These steps can be implemented using the following approach:
The first step will produce a value e that is either 1 or p-1. The second step is so fast compared to the first that that its dependence on e is likely to not matter; implementers can also ensure that it runs in the same amount of time regardless of e.
If having this algorithm run in constant time is not important, then there are much faster algorithms to compute the Legendre symbol (which is the same as the Jacobi symbol because p is a prime). See, for example, Section 12.3 of [ntb].
[TODO: In addition to Elligator, it would be nice to have a generic ECVRF_hash_to_curve algorithm that runs in constant time and provides a generic way to hash an octet string onto any elliptic curve. There is an an upcoming draft that deals with this [SW18]. Some other useful algorithms include [Icart09], and Shallue-Woestijne-Ulas algorithm from [BCIMRT10].]
ECVRF_hash_points(p_1, p_2, ..., p_j)
Input:
Output:
Steps:
ECVRF_decode_proof(pi)
Input:
Output:
Steps:
This document defines ECVRF-P256-SHA256 as follows:
This document defines ECVRF-ED25519-SHA512 as follows:
This document defines ECVRF-ED25519-SHA512-Elligator2 as follows:
The ECVRF as specified above is a VRF that satisfies the "trusted uniqueness", "trusted collision resistance", and "full pseudorandomness" properties defined in Section 3. In order to obtain "full uniqueness" and "full collision resistance" (which provide protection against a malicious VRF public key), the Verifier MUST perform the following additional validation procedure upon receipt of the public VRF key. The public VRF key MUST NOT be used if this procedure returns "INVALID".
Note that this procedure is not sufficient if the elliptic curve E or the point g, the generator of group G, is untrusted. If the prover is untrusted, the Verifier MUST obtain E and g from a trusted source, such as a ciphersuite specification, rather than from the prover.
This procedure supposes that the public key provided to the Verifier is an octet string. The procedure returns "INVALID" if the public key in invalid. Otherwise, it returns y, the public key as an EC point.
ECVRF_validate_key(PK)
Input:
Output:
Steps:
An implementation of the RSA-FDH-VRF (SHA-256) and ECVRF-P256-SHA256 was first developed as a part of the NSEC5 project [I-D.vcelak-nsec5] and is available at <http://github.com/fcelda/nsec5-crypto>. The ECVRF implementation may be out of date as this spec has evolved.
The Key Transparency project at Google uses a VRF implemention that is similar to the ECVRF-P256-SHA256, with a few minor changes including the use of SHA-512 instead of SHA-256. Its implementation is available <https://github.com/google/keytransparency/blob/master/core/vrf/vrf.go>
An implementation by Yahoo! similar to the ECVRF is available at <https://github.com/r2ishiguro/vrf>.
An implementation similar to ECVRF is available as part of the CONIKS implementation in Golang at <https://github.com/coniks-sys/coniks-go/tree/master/crypto/vrf>.
Open Whisper Systems also uses a VRF very similar to ECVRF-ED25519-SHA512-Elligator, called VXEdDSA, and specified here: <https://whispersystems.org/docs/specifications/xeddsa/>
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 Section 5.1-Section 5.5 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 Section 5.6) 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.
[nsec5ecc] 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.
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 Section 5.5 are designed with this requirement in mind.
The ECVRF_hash_to_curve_try_and_increment 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 algorithm's running time depends on the VRF input alpha, and thus creates a timing channel that can be used to learn information about alpha. That said, for most inputs the amount of information obtained from such a timing attack is likely to be small (1 bit, on average), since the algorithm is expected to find a valid curve point after only two attempts. However, there might be inputs which cause the algorithm to make many attempts before it finds a valid curve point; for such inputs, the information leaked in a timing attack will be more than 1 bit.
Meanwhile, ECVRF-ED25519-SHA512-Elligator2 runs in constant time if the implementation of the ECVRF_hash_to_curve function specified in Section 5.4.1.2 also runs in constant time.
The VRF proof pi is not designed to provide secrecy and, in general, may reveal the VRF input alpha. Anyone who knows PK and pi is able to perform an offline dictionary attack to search for alpha, by verifying guesses for alpha using VRF_verify. This is in contrast to the VRF hash output beta which, without the proof, is pseudorandom and thus is designed to reveal no information about alpha.
The VRFs specified in this document allow for read-once access to the input alpha for both signing and verifying. Thus, additional prehashing of alpha (as specified, for example, in [RFC8032] for EdDSA signatures) is not needed, even for applications that need to handle long alpha or to support the Initialized-Update-Finalize (IUF) interface (in such an interface, alpha is not supplied all at once, but rather in pieces by a sequence of calls to Update). The ECVRF, in particular, uses alpha only in ECVRF_hash_to_curve. The curve point h becomes the representative of alpha thereafter. Note that the suite octet and the public key are hashed together with alpha in ECVRF_hash_to_curve, which ensures that the curve (including the generator g) and the public key are included indirectly into subsequent hashes.
Hashing is used for four different purposes in ECVRF (namely, in hash_to_curve, nonce_generation, hash_points, and proof2hash). The theoretical analysis assumes each of these four functions is a separate random oracle. This analysis still holds even if the same hash function is used, as long as the four queries made to the hash function for a given SK and alpha are overwhelmingly unlikely to equal each other or to any queries made to the hash function for the same SK and different alpha. This is indeed the case for the ECVRF ciphersuites defined in this document, because:
If future designs need to specify variants (e.g., additional ciphersuites) of the design in this document, then, to avoid the possibility that an adversary can obtain a VRF output under one variant, and then claim it was obtained under another variant, they should specify a different suite constant. This way, the inputs to the hash_to_curve hash function used in producing h are guaranteed to be different; since all the other hashing done by the prover depends on h, inputs all the hash functions used by the prover will also be different as long as hash_to_curve is collision resistant.
Note to RFC Editor: if this document does not obsolete an existing RFC, please remove this appendix before publication as an RFC.
This document also would not be possible without the work of Moni Naor (Weizmann Institute), Sachin Vasant (Cisco Systems), and Asaf Ziv (Facebook). Shumon Huque (Salesforce), David C. Lawerence (Akamai), and Trevor Perrin provided valuable input to this draft.
| [BCIMRT10] | Brier, E., Coron, J., Icart, T., Madore, D., Randriam, H. and M. Tibouchi, "Efficient Indifferentiable Hashing into Ordinary Elliptic Curves", in CRYPTO, 2010. |
| [I-D.vcelak-nsec5] | Vcelak, J., Goldberg, S., Papadopoulos, D., Huque, S. and D. Lawrence, "NSEC5, DNSSEC Authenticated Denial of Existence", Internet-Draft draft-vcelak-nsec5-07, June 2018. |
| [Icart09] | Icart, T., "How to Hash into Elliptic Curves", in CRYPTO, 2009. |
| [MRV99] | Michali, S., Rabin, M. and S. Vadhan, "Verifiable Random Functions", in FOCS, 1999. |
| [nsec5ecc] | Papadopoulos, D., Wessels, D., Huque, S., Vcelak, J., Naor, M., Reyzin, L. and S. Goldberg, "Making NSEC5 Practical for DNSSEC", in ePrint Cryptology Archive 2017/099, February 2017. |
| [ntb] | Shoup, V., "A Computational Introduction to Number Theory and Algebra", 2008. |
| [SW18] | Sullivan, E. and C. Wood, "Hashing to Elliptic Curves", 2018. |