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Pollard's ''p'' − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with specific types of factors; it is the simplest example of an algebraic-group factorisation algorithm. The factors it finds are ones for which the number preceding the factor, ''p'' − 1, is powersmooth; the essential observation is that, by working in the multiplicative group modulo a composite number ''N'', we are also working in the multiplicative groups modulo all of ''Ns factors. The existence of this algorithm leads to the concept of safe primes, being primes for which ''p'' − 1 is two times a Sophie Germain prime ''q'' and thus minimally smooth. These primes are sometimes construed as "safe for cryptographic purposes", but they might be ''unsafe'' — in current recommendations for cryptographic strong primes (''e.g.'' ANSI X9.31), it is necessary but not sufficient that ''p'' − 1 has at least one large prime factor. Most sufficiently large primes are strong; if a prime used for cryptographic purposes turns out to be non-strong, it is much more likely to be through malice than through an accident of random number generation. This terminology is considered obsolete by the cryptography industry. () ==Base concepts== Let ''n'' be a composite integer with prime factor ''p''. By Fermat's little theorem, we know that for all integers ''a'' coprime to ''p'' and for all positive integers ''K'': : If a number ''x'' is congruent to 1 modulo a factor of ''n'', then the will be divisible by that factor. The idea is to make the exponent a large multiple of ''p'' − 1 by making it a number with very many prime factors; generally, we take the product of all prime powers less than some limit ''B''. Start with a random ''x'', and repeatedly replace it by as ''w'' runs through those prime powers. Check at each stage, or once at the end if you prefer, whether is not equal to 1. 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「Pollard's p − 1 algorithm」の詳細全文を読む スポンサード リンク
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