Find $F(2x)$ from $F(x) = a^x \bmod p$ Given $F(x) = a^x \bmod p$, where $a$ is a primitive root of $p$,
Is it possible to work out what $F(2x)$ or $F(3x)$, etc if you know
what $F(x)$ is but not $x$.

If you use $F(x)$ then $F(2x)$, etc as session keys for a cryptosystem would be knowing a previous key compromise future keys.
Rephrasing fkraiem's comment as an answer:


  Hint:   $$a^{2x}=(a^x)^2$$


which also holds $\bmod p$.