# Discrete logarithm

## The problem of inverting exponentiation in finite groups / From Wikipedia, the free encyclopedia

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In mathematics, for given real numbers *a* and *b*, the logarithm log_{b} *a* is a number *x* such that *b*^{x} = *a*. Analogously, in any group *G*, powers *b*^{k} can be defined for all integers *k*, and the **discrete logarithm** log_{b} *a* is an integer *k* such that *b*^{k} = *a*. In number theory, the more commonly used term is **index**: we can write *x* = ind_{r} *a* (mod *m*) (read "the index of *a* to the base *r* modulo *m*") for *r*^{x} ≡ *a* (mod *m*) if *r* is a primitive root of *m* and gcd(*a*,*m*) = 1.

Discrete logarithms are quickly computable in a few special cases. However, no efficient method is known for computing them in general. Several important algorithms in public-key cryptography, such as ElGamal base their security on the assumption that the discrete logarithm problem over carefully chosen groups has no efficient solution.[1]