I am new to modular arithmetic. I have gone through a few formulae related to this but I am unable to solve problems like (5^12) mod 23. I am not really bothered about the final answer, but mainly on the approach on these kinds of problems (or maybe more involving complex powers). Any help is appreciated. Thank you.
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1Please read the following: https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic/a/fast-modular-exponentiation – Noble Mushtak Dec 29 '18 at 00:57
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Fermat's little theorem and Euler's totient function come to mind. – For the love of maths Dec 29 '18 at 00:58
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- Rewriting "big" numbers like $920 \,\rm{mod}\, 1000$ as "smaller" numbers: $920 \equiv -80$.
- Finding patterns in exponents (for example, $2^n \,\rm{mod}\, 10$ has a cycle $2,\, 4,\, 8,\, 6$ with $4$ steps).
- Reducing big exponents using Euler's $\phi$.
- Squaring (!). For example, in $231^{19} \,\rm{mod}\,517$, you might find, first, $231^2$ and then use that $231^{19} = {(231^2)}^9\times 231$. Iterating this makes every congruence so, oh so much easier..
Those are the main techniques I'm aware of.
Lucas Henrique
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