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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.

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    $\begingroup$ Hint: $a^{2x} = (a^x)^2$. $\endgroup$ – fkraiem Oct 15 '18 at 3:08
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Rephrasing fkraiem's comment as an answer:

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

which also holds $\bmod p$.

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    $\begingroup$ FWIW, when I write a hint as a comment like that, it's with the expectation that OP will expand it into a full solution and submit it themselves (a good way to get rep!). Maybe I should make that explicit... $\endgroup$ – fkraiem Oct 15 '18 at 17:58

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