Q.4. (Marks:20) If ElGamal signature is generated using following steps.
1. Key generation: secret key a with 1<a<p-2
2. Public key, $\beta = \alpha^a \mod p$; $\alpha$,p are also public.
3. p→ large prime number and $\alpha$→generator of $Z_p^*$
4. Calculate signature;
a. repeat
i. Choose random k such that 1 <k<p-1 and gcd(k, p-1) = 1.
ii. $r = \alpha^k \mod p$ and $s = (m-ar)k^{-1} \mod (p-1)$
b. until $s \ne 0$;
At the end of above process, (r,s) represent signature associated with the document
m. The signature is verified by checking if $\beta^r \equiv \alpha^m \mod p$. Prove that (assuming all
steps in algorithm are done properly) $\beta^{rs}$ is infact congruent to $\alpha^m \mod p$.