Discrete logarithm problem
In the mathematics of the real numbers, the logarithm logba is a number x such that bx = a, for given numbers a and b. Analogously, in any group G, powers bk can be defined for all integers k, and the discrete logarithm logba is an integer k such that bk = a.
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 base their security on the assumption that the discrete logarithm problem over carefully chosen groups has no efficient solution.
Let G be any group. Denote its group operation by multiplication and its identity element by 1. Let b be any element of G. For any positive integer k, the expression bk denotes the product of b with itself k times:
Similarly, let b-k denote the product of b−1 with itself k times. For k = 0, the kth power is the identity: b0 = 1.
Let a also be an element of G. An integer k that solves the equation bk = a is termed a discrete logarithm (or simply logarithm, in this context) of a to the base b. One writes k = logba.
The powers of 10 form an infinite subset G = {…, 0.001, 0.01, 0.1, 1, 10, 100, 1000, …} of the rational numbers. This set G is a cyclic group under multiplication, and 10 is a generator. For any element a of the group, one can compute log10a. For example, log10 10000 = 4, and log10 0.001 = −3. These are instances of the discrete logarithm problem.
Other base-10 logarithms in the real numbers are not instances of the discrete logarithm problem, because they involve non-integer exponents. For example, the equation log10 53 = 1.724276… means that 101.724276… = 53. While integer exponents can be defined in any group using products and inverses, arbitrary real exponents in the real numbers require other concepts such as the exponential function.
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