Logarithm Problem Solving

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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. 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. 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.

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. 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. 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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To create this article, volunteer authors worked to edit and improve it over time. However none of them run in polynomial time (in the number of digits in the size of the group).becomes a product bk, and equality means congruence modulo p in the integers. While computing discrete logarithms and factoring integers are distinct problems, they share some properties: ) there is not only no efficient algorithm known for the worst case, but the average-case complexity can be shown to be about as hard as the worst case using random self-reducibility.By precomputing these three steps for a specific group, one need only carry out the last step, which is much less computationally expensive than the first three, to obtain a specific logarithm in that group.It turns out that much Internet traffic uses one of a handful of groups that are of order 1024 bits or less, e.g.Discrete logarithms are quickly computable in a few special cases.However, no efficient method is known for computing them in general.A general algorithm for computing log a in finite groups G is to raise b to larger and larger powers k until the desired a is found.This algorithm is sometimes called trial multiplication.In the special case where b is the identity element 1 of the group G, the discrete logarithm log The discrete logarithm problem is considered to be computationally intractable.That is, no efficient classical algorithm is known for computing discrete logarithms in general.

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