# 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 knowing a previous key compromise future keys.

• Hint: $a^{2x} = (a^x)^2$. – fkraiem Oct 15 '18 at 3:08

Hint: $$a^{2x}=(a^x)^2$$
which also holds $$\bmod p$$.