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The Solovay–Strassen primality test, developed by Robert M. Solovay and Volker Strassen, is a probabilistic test to determine if a number is composite or probably prime. It has been largely superseded by the Baillie-PSW primality test and the Miller–Rabin primality test, but has great historical importance in showing the practical feasibility of the RSA cryptosystem. ==Concepts== Euler proved〔Euler's criterion〕 that for any prime number ''p'' and any integer ''a'', : where is the Legendre symbol. The Jacobi symbol is a generalisation of the Legendre symbol to , where ''n'' can be any odd integer. The Jacobi symbol can be computed in time O((log ''n'')²) using Jacobi's generalization of law of quadratic reciprocity. Given an odd number ''n'' we can contemplate whether or not the congruence : holds for various values of the "base" ''a'', given that ''a'' is relatively prime to ''n''. If ''n'' is prime then this congruence is true for all ''a''. So if we pick values of ''a'' at random and test the congruence, then as soon as we find an ''a'' which doesn't fit the congruence we know that ''n'' is not prime (but this does not tell us a nontrivial factorization of ''n''). This base ''a'' is called an ''Euler witness'' for ''n''; it is a witness for the compositeness of ''n''. The base ''a'' is called an ''Euler liar'' for ''n'' if the congruence is true while ''n'' is composite. For every composite odd ''n'' at least half of all bases : are (Euler) witnesses:〔(PlanetMath )〕 this contrasts with the Fermat primality test, for which the proportion of witnesses may be much smaller. Therefore, there are no (odd) composite ''n'' without many witnesses, unlike the case of Carmichael numbers for Fermat's test. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Solovay–Strassen primality test」の詳細全文を読む スポンサード リンク
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