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If values $a$, $b$, and $c$ are known, is there an efficient way to find $x$ in the equation: $a^{x} \bmod b = c$?

E.g. finding $x$ in $128^{x}\bmod 209 = 39$.

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2 Answers 2

up vote 5 down vote accepted

This is the Discrete Logarithm Problem. If by "efficient" you mean "polynomial time general purpose algorithm", the answer is that none are known.

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A better reference than Wikipedia for the discrete logarithm problem is Andrew Sutherland's 2007 MIT Thesis Order Computations in Generic Groups. Here's an excerpt from p. 14 that provides a concise summary of the current state of knowledge.

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