I'm trying to solve excersise 7.6 from Hoffstein - Introduction to Mathematical Cryptography page 459 (hhttp://goo.gl/oRyInT)
Let $p$ be a prime and let $i$ and $j$ be integers with $gcd(j, p − 1) = 1$. Set $S_1 \equiv g^i v^j \mod p$, $S_2 \equiv −S_1 j^{-1} \mod p - 1$, $D \equiv −S_1 i j^{−1} \mod p − 1$.
I need to prove that ($S_1$, $S_2$) is a valid ElGamal signature on the document $D$ and for the verification key $v$. What do I need to prove this? Is it enough to show soundnes of such variation of ElGamal signature?
We need to check equality $(y^a)*(a^b) = q^w$ holds.
$(y^a)*(q^i * y^j) ^ {-aj^{-1}} = y^{a - a*j*j^{-1}} * q^{-aij^{-1}}$
Using Eulers theorem $j*j^{-1} = 1$
Then $q^{-aij^{-1}} = q^w$, then $q^w = q^w$