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:

$ExpRand_{VRF}^{\mathcal{A}}(1^\lambda)$:

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

$ExpCol_{VRF}^\mathcal{A}(1^\lambda)$:

$(pk,sk) \leftarrow \mathcal{A_1}(\mathsf{Gen}(1^{\lambda}))$
$(X,X',st) \leftarrow \mathcal{A_2}(sk,pk)$
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.