VRF Security Properties

We need a VRF to satisfy the following properties:

Correctness: If \((Y,\pi)=Eval(sk,X)\) then \(Verify(pk,X,Y,\pi)=1\)

Uniqueness: There do not exist tuples \((Y,\pi)\) and \(Y',\pi'\) with \(Y \ne Y'\) and: \(\mathsf{Verify}(pk,X,Y,\pi)=\mathsf{Verify}(pk,X,Y',\pi')=1\)

Pseudorandomess: For any adversary \(\mathcal{A}\) the probability \(|Pr[ExpRand^A_{VRF}(\lambda)=1]-\dfrac{1}{2}|\) is negilible where \(ExpRand_{VRF}^{\mathcal{A}}(1^\lambda)\) is defined as follows:


  • \((sk,pk) \leftarrow \mathsf{Gen}(1^{\lambda})\)
  • \((X^*,st) \leftarrow \mathcal{A}^{\mathcal{O_{VRF}}(.)}(pk)\)
  • \(Y_0 \leftarrow \mathsf{Eval}(X*,sk)\)
  • \(Y_1 \leftarrow \{0,1\}^{out(\lambda)}\)
  • \(\{0,1\} {\stackrel{$}{\leftarrow}} b\)
  • \(b' \leftarrow \mathcal{A}(Y_b,st)\)
  • Return \(b=b'\)

The oracle \(\mathcal{O_{VRF}}(.)\) works as follow: Given an input \(x\), it outputs the VRF value computed by \(x\) and its proof.

In the paper of [PWHVNRG17], the authors stated that a VRF must also be collision resistant. This property is formally defined below:

Collision Resistant: Collision Resistant: For any adversarial prover \(\mathcal{A}=(\mathcal{A_1},\mathcal{A_2})\) the probability \(Pr\left[ExpCol_{VRF}^\mathcal{A}(1^\lambda)=1\right]\) is negligible where \(ExpCol_{VRF}^\mathcal{A}(1^\lambda)\) is defined as follows:


  • \((pk,sk) \leftarrow \mathcal{A_1}(\mathsf{Gen}(1^{\lambda}))\)
  • \((X,X',st) \leftarrow \mathcal{A_2}(sk,pk)\)
  • Return \(X \ne X'\) and \(\mathsf{Eval}(X,sk)=\mathsf{Eval}(X',sk)\)

It is interesting to see that, VRF can be used for signing messages. However, several signing algorithms such as ECDSA cannot be used to build a VRF. For a given message and a secret key, there can be multiple valid signatures, thus an adversiral prover could produce different valid outputs from a given input, and chooses the one that benefits him. This contradicts the uniqueness property of VRF.