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| {{Unreferenced|date=November 2006}}
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| In [[computational number theory]], the '''factor base''' is a mathematical tool commonly used in algorithms involving extensive [[sieve theory|sieving]] of potential factors.
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| ==Usage==
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| The factor base is a relatively small [[set theory|set]] of [[prime number]]s ''P''. Say we want to factorize an integer ''n''. We generate, in some way, a large number of [[Modular arithmetic#Congruence_relation|congruent]] integer pairs (''x'', ''y'') for which <math>x \neq y</math> and <math>\textstyle x \equiv y \pmod{n}</math>, and ''x'', ''y'' can be completely factorized over the chosen factor base—that is, they are [[smooth number|''p''-smooth]] for the largest prime ''p'' in ''P''.
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| We find a subset ''S'' of the integer pairs such that <math>\textstyle \prod_{} x \hbox{, } (x \hbox{ in } S)</math> and <math>\textstyle \prod_{} y \hbox{, } (y \hbox{ in } S)</math> are both [[square number|perfect square]]s. Over our factor base this reduces to adding [[prime signature|exponents of their prime factors]], [[GF(2)|modulo 2]], as we can distinguish squares from non-squares simply by checking the [[parity (mathematics)|parity]] of the exponents.
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| We can represent each ''x'' and ''y'' as a [[vector space|vector]] of a [[matrix (mathematics)|matrix]] with 0 and 1 entries for the parity of each exponent. This essentially reformulates the problem into a [[system of linear equations]], which can be solved using numerous methods such as [[Gaussian elimination]]; in practice advanced methods like the [[block Lanczos algorithm for nullspace of a matrix over a finite field|block Lanczos algorithm]] are used, that take advantage of certain properties of the system.
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| Once such a subset is found, we have essentially found a [[congruence of squares]] modulo ''n'' and can attempt to factorize ''n''. This congruence may generate the trivial <math>\textstyle n = 1 \cdot n</math>; in this case we try to find another suitable subset. If no such subset is found, we can search for more (''x'', ''y'') pairs, or try again using a different factor base.
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| ==Algorithms==
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| Factor bases are used in, for example, [[Dixon's factorization method|Dixon's factorization]], the [[quadratic sieve]], and the [[general number field sieve|number field sieve]]. The difference between these algorithms is essentially the methods used to generate (''x'', ''y'') candidates.
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| {{DEFAULTSORT:Factor Base}}
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| [[Category:Integer factorization algorithms]]
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| {{Numtheory-stub}}
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