Evaluate hash function I am trying to figure out how to evaluate the security of a hash function. If I have defined a candidate hash function: $$H(x) = α^x\text{ mod }p$$ how do I then evaluate it with respect to collision attacks, second preimage attacks, and preimage attacks? Hope someone can help
If $x=y \mod \phi(p)$, then $H(x)= H(y)$. Then it doesn't seem collision resistant.
For the same reason not second preimage attacks. $H(x + \phi(p))= H(y)$.
(If $p$ is prime $\phi(p)=p-1$, thus it's easy to compute it).
About the preimage attack, it's completely equivalent to discrete logarithm in $\mathbb{Z}^{\star}_p$, then you have to be carrefull about the choice of $p$ (basically $p-1$ should have a huge factor).