Given a prime $p$ and number $a$ isn't divided by $p$, how do I find the smallest positive integer number $n$ such that $a^n≡1\pmod p$?
Example for $a=10; p=13$; $n$ would be $6$.
Is there an algorithms that faster than brute-force with $O(p)$?
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Is this for cryptography? – Full Array Jan 12 '24 at 21:44
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https://en.wikipedia.org/wiki/Discrete_logarithm – Arthur Jan 12 '24 at 21:53
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@Arthur That's for general equations of the form $a^n \equiv b$, but surely something can be said for the specific case that $b = 1$. – Ben Grossmann Jan 12 '24 at 22:32
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1Maybe, Euler's theorem can help (see https://en.wikipedia.org/wiki/Euler%27s_theorem) or, even better, you can consider Carmichael's function. Another very powerful tool is Zsigmondy's theorem... – Marco Ripà Jan 12 '24 at 22:38