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Suppose $p$ is prime, and $g, a\in\mathbb F_p$ are given elements with $g$ a primitive root. The discrete log problem poses the task of finding an integer $x$ such that $g^x=a$. Show that even if $x$ cannot be recovered, one can check if $x$ is odd or even by inspecting the element $a^{(p-1)/2}$.

I'm not sure how to approach this question. I tried $g^{x(p-1)/2} = a^{(p-1)/2} = 1$.

Any help would be appreciated! Thanks.

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  • $\begingroup$ What do you mean by "tried"? The equation $a^{(p-1)/2}\overset{\text?}=1$ does not necessarily hold. $\endgroup$
    – yyyyyyy
    Feb 8 '15 at 22:34
  • $\begingroup$ In particular, what is $g^{x(p-1)/2}$ if $x$ is even? What is it if it is odd? $\endgroup$
    – poncho
    Feb 8 '15 at 22:35
  • $\begingroup$ The question is strongly connected with Legendre Symbol of a, as you observe it. If Legendre symbol of a is +1, then x is even! $\endgroup$ Feb 8 '15 at 22:45
  • $\begingroup$ if x is even then 2 divides it but I don't know what that means. does it imply that it's equal to 1? by fermat's little theorem $\endgroup$ Feb 8 '15 at 22:47
  • $\begingroup$ What happens when x is odd? $\endgroup$ Feb 8 '15 at 23:12
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Claim. $a^{(p-1)/2} = 1$ if and only if $x$ is even.

Proof. If $x$ is even, let $x = 2y$. Then $$a^{(p-1)/2} = (g^x)^{(p-1)/2} = g^{2y(p-1)/2} = (g^{p-1})^y = 1^y = 1.$$

If $x$ is odd, let $x = 2y+1$. Then $$a^{(p-1)/2} = g^{(2y+1)(p-1)/2} = \dots$$ (remember here that $g$ is a generator of $\mathbf{F}_p^*$).

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  • $\begingroup$ If x is odd, we are left with $g^{(p-1)/2}$. However i fail to understand what that is equal to. I know $g^{(p-1)} = 1$ since $g$ is a generator $\endgroup$ Feb 9 '15 at 0:06
  • $\begingroup$ You don't care what it is equal to. ;) What is important is what it is not equal to. $\endgroup$
    – fkraiem
    Feb 9 '15 at 0:12
  • $\begingroup$ By the way, $g^{p-1} = 1$ is true for any $g$. The fact that $g$ is a generator implies something more than that. $\endgroup$
    – fkraiem
    Feb 9 '15 at 0:17
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    $\begingroup$ @Bren2010 What are you talking about? We are working in the full group $\mathbf{F}_p^*$, which in your example has order $10$, not $5$. $\endgroup$
    – fkraiem
    Feb 9 '15 at 0:29
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    $\begingroup$ @mikerussel Oh and by the way, the fact that $a^{(p-1)/2} = -1$ if $x$ is odd follows simply from what you have found. You have $a^{(p-1)/2} = g^{(p-1)/2}$, so $(a^{(p-1)/2})^2 = g^{p-1} = 1$, meaning that $a^{(p-1)/2}$ is a root of the polynomial $X^2-1$. Since we are over a field, this polynomial has at most two roots, which are easily seen to be $1$ and $-1$. Since $a^{(p-1)/2}$ is not $1$, it must be $-1$. $\endgroup$
    – fkraiem
    Feb 9 '15 at 0:42

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