Verifiable random function

In cryptography, a verifiable random function (VRF) is a pseudo-random function that provides publicly verifiable proofs of its outputs' correctness. Given an input value x, the owner of the secret key SK can compute the function value y = F_SK(x) and the proof p_SK(x). Using the proof and the public key PK (sometimes known as a verification key and denoted VK), everyone can check that the value y = F_SK(x) was indeed computed correctly, yet this information cannot be used to find the secret key. A verifiable random function can be viewed as a public-key analogue of a keyed cryptographic hash and as a cryptographic commitment to an exponentisally large number of seemingly random bits. The concept of a verifiable random function is closely related to that of a verifiable unpredictable function (VUF), whose outputs are hard to predict but do not necessarily seem random.

The concept of a VRF was introduced by Micali, Rabin, and Vadhan in 1999. Since then, verifiable random functions have found widespread use in cryptocurrencies, as well as for proposals in protocol design and cybersecurity.

Constructions

In 1999, Micali, Rabin, and Vadhan introduced the concept of a VRF and proposed the first such one. The original construction was rather inefficient: it first produces a verifiable unpredictable function, then uses a hard-core bit to transform it into a VRF; moreover, the inputs have to be mapped to primes in a complicated manner: namely, by using a prime sequence generator that generates primes with overwhelming probability. The verifiable unpredictable function thus proposed, which is provably secure if a variant of the RSA problem is hard, is defined as follows: The public key PK is $(m,r,Q,coins)$, where m is the product of two random primes, r is a number randomly selected from $\mathbb {Z} _{r}^{*}$, coins is a randomly selected set of bits, and Q a function selected randomly from all polynomials of degree $2k^{2}-1$ over the field $GF(2^{k})$. The secret key is $(PK,\phi (m))$. Given an input x and a secret key SK, the VUF uses the prime sequence generator to pick a corresponding prime $p_{x}$ (the generator requires auxiliary inputs Q and coins), and then computes and outputs $r^{1/p_{x}}{\pmod {m}}$, which is easily done by knowledge of $\phi (m)$.

Later, in 2005, an efficient and practical verifiable random function was proposed by Dodis and Yampolskiy. When the input $x$ is from a small domain (the authors then extend it to a larger domain), the function can be defined as follows:

$F_{SK}(x)=e(g,g)^{1/(x+SK)}\quad {\mbox{and}}\quad p_{SK}(x)=g^{1/(x+SK)},$

where e(·,·) is a bilinear map. To verify whether $F_{SK}(x)$ was computed correctly or not, one can check if $e(g^{x}PK,p_{SK}(x))=e(g,g)$ and $e(g,p_{SK}(x))=F_{SK}(x)$. To extend this to a larger domain, the authors use a tree construction and a universal hash function. This is secure if it is hard to break the "q-Diffie-Helman inversion assumption", which states that no algorithm given $(g,g^{x},\dots ,g^{x^{q}})$ can compute $g^{1/x}$, and the "q-decisional bilinear Diffie-Helman inversion assumption", which states that it is impossible for an efficient algorithm given $(g,g^{x},\ldots ,g^{(x^{q})},R)$ as input to distinguish $R=e(g,g)^{1/x}$ from random, in the group $\mathbb {G}$.

In 2015, Hofheinz and Jager constructed a VRF which is provably secure given any member of the "(n − 1)-linear assumption family", which includes the decision linear assumption. This is the first such VRF constructed that does not depend on a "Q-type complexity assumption".

In 2019, Bitansky showed that VRFs exist if non-interactive witness-indistinguishable proofs (that is, weaker versions of non-interactive zero-knowledge proofs for NP problems that only hide the witness that the prover uses), non-interactive cryptographic commitments, and single-key constrained pseudorandom functions (that is, pseudorandom functions that only allow the user to evaluate the function with a preset constrained subset of possible inputs) also do.

In 2020, Esgin et al. proposed a post-quantum secure VRF based on lattice-based cryptography.

Uses and applications

VRFs provide deterministic precommitments which can be revealed at a later time using proofs which can only be generated by a private key. This is useful for providing a 1:1 mapping of low entropy inputs (e.g. names, email addresses, phone numbers) to some random values which can be committed to in advance, e.g. through a timestamping service such as a transparency log.

Unlike traditional digital signature algorithms, VRF outputs can be published publicly without being subject to a preimage attack, even if the verifier knows the public key (but not the proof). This is useful to prevent enumeration of the names/identifiers in a directory which is using a transparency system.

In protocol design

VRFs have been used to make:

• Resettable zero-knowledge proofs with three rounds (in the bare model)
• A non-interactive lottery system
• A verifiable transaction escrow scheme

In Internet security

DNSSEC is a system that prevents attackers from tampering with Domain Name System messages, but also suffers from the vulnerability of zone enumeration. The proposed NSEC5 system, which uses VRFs, provably prevents this kind of attack.

In cryptocurrency

Cardano and Polkadot implement VRFs for block production.

The Internet Computer cryptocurrency uses a VRF to produce a decentralized random beacon whose output is unpredictable to anyone until they become available to everyone. More precisely, its protocol allows clients to agree on a VRF (i.e. a commitment to a deterministic pseudorandom sequence) and to produce one new output thereof every round by using threshold signatures and distributed key generation.

Algorand uses VRFs to perform cryptographic sortition. In this platform, every block produces a new random selection seed, and a user secretly checks whether they were selected to participate in the consensus protocol by evaluating a VRF with their secret participation key and the selection seed. (This also produces a proof, which the user can send to anyone to show that they have been selected to participate.) Thusly, accounts are selected to propose blocks for a given round. The particular VRF that Algorand uses is based on elliptic-curve cryptography, specifically Curve25519; proposed by Sharon Goldberg, Moni Naor, Dimitris Papadopoulos, Leonid Reyzin, and Jan Včelák, the function is currently undergoing standardization. Algorand's own implementation of this function has been open-source since October 2018.

VRFs based on elliptic curve cryptography have been implemented in the programming language Solidity.

In May 2020, Chainlink announced that it launched Chainlink VRF, a service that uses verifiable random functions to generate verifiable randomness on-chain. To use Chainlink VRF, a smart contract supplies a seed (which should be unpredictable to the oracles to whom it is provided), and the seed in turn is used to generate a random number that is sent back to the contract; this is published on-chain along with a proof and verified using the oracle's public key and the application's seed. Each oracle, when generating randomness, uses its own secret key.

In July 2021, Harmony, a cryptocurrency project that bridges between blockchains, announced that it had introduced VRFs. The same month, Oraichain, a cryptocurrency project involving [[Artificial intelligence|artificial intelligence]], made the same kind of announcement.