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Thread: How to solve a^x mod b = c?

  1. #1
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    How to solve a^x mod b = c?

    How to solve $\displaystyle a^x$ $\displaystyle mod$ $\displaystyle b = c$ for $\displaystyle x$ minimum integer solution.

    With known integers $\displaystyle a$,$\displaystyle b$, and $\displaystyle c$.
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  2. #2
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    Re: How to solve a^x mod b = c?

    If $\displaystyle b=1,2,4,p^\alpha,2p^\alpha$ with $\displaystyle \alpha\ge 1$ and $\displaystyle p$ an odd prime, then $\displaystyle b$ has a primitive root $\displaystyle g \mod b$.
    If $\displaystyle (a,b)=(c,b)=1$ then $\displaystyle x $ind$\displaystyle _g a\equiv $ ind$\displaystyle _g c \pmod {\phi(b)}$ a linear congruence in $\displaystyle x$.
    If $\displaystyle d=($ind$\displaystyle _g a,\phi(b))|$ ind$\displaystyle _g c$ then there are exactly $\displaystyle d$ solutions.

    Then start by factoring $\displaystyle b$. If $\displaystyle b$ has a factor $\displaystyle 2^f$ with $\displaystyle f\ge 3$ then solve the equation $\displaystyle \mod 2^f$ first.
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