# Security analysis for a extended ElGamal Signature against selective unforgeability

Let $H$ is a cryptography hash function and $\Pi=(\mathsf{G}, \mathsf{S}, \mathsf{V})$ is a digital signature, as follows:

$(h_1=g^x,h_2=g^y) \leftarrow \mathsf{G}(1^n)$, where $x,y$ uniformly random from $\mathbb{Z}^*_q \ .$

$(r=g^k,s=(H(m)-x \cdot r)\cdot k^{-1},z= y^{-1} \cdot k ) \leftarrow \mathsf{s}_{x,y}(m)$, where $k$ uniformly random from $\mathbb{Z}^*_q \ .$

$b:=\mathsf{V}_{h_1,h_2}(m,(r,s,z))$, where $b := \begin{cases} 1 & \text{if } (g^{H(m)} = h_1^r \cdot h_2^{z \cdot s}) \\ 0 & \text{otherwise} \end{cases}$ The above construction is similar to ElGamal signature (https://en.wikipedia.org/wiki/ElGamal_signature_scheme).

1- Where can I find proof of existential unforgeability ElGamal signature?

2- Does the above construction is a secure digital signature against existential unforgeability?