en>Qwertyus: I don't see why PARI/GP deserves special mention, in the lede

2014-09-23T21:36:39Z

I don't see why PARI/GP deserves special mention, in the lede

en>Xqbot: Robot: ru:Алгоритм Берлекэмпа is a good article

2013-12-31T11:39:57Z

Robot: ru:Алгоритм Берлекэмпа is a good article

← Older revision		Revision as of 13:39, 31 December 2013
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	~~Hello~~. ~~Allow me introduce~~ the ~~writer~~. ~~Her name~~ is ~~Emilia Shroyer but it~~'s ~~not~~ the ~~most feminine name out there~~. ~~To collect coins~~ is 1 of the ~~things I adore most~~. ~~California~~ is ~~our birth location~~. ~~He used~~ to ~~be unemployed but~~ now he is a ~~computer operator but his marketing never comes~~.<br><br>~~Visit my web~~-~~site~~ - ~~at home std testing (~~[http://~~test~~.~~ithink~~-~~now~~.~~org~~/~~content~~/~~think~~-~~f83d96d08fe71e25bf80a6cb59048a18 browse around this website~~])		In [[Computational mathematics\|computational]] [[Abstract algebra\|algebra]], the '''Cantor–Zassenhaus algorithm''' is a well known method for factorising [[polynomial]]s over [[finite field]]s (also called Galois fields).

			The algorithm consists mainly of exponentiation and polynomial [[greatest common divisor\|GCD]] computations. It was invented by D. Cantor and [[Hans Zassenhaus]] in 1981.

			It is arguably the dominant algorithm for solving the problem, having replaced the earlier [[Berlekamp's algorithm]] of 1967. It is currently implemented in many well-known [[computer algebra system]]s, including [[PARI/GP]].

			==Overview==

			===Background===
			The Cantor–Zassenhaus algorithm takes as input a squarefree polynomial <math>f(x)</math> (i.e. one with no repeated factors) of degree ''n'' with coefficients in a finite field <math>\mathbb{F}_q</math> whose [[irreducible polynomial]] factors are all of equal degree (algorithms exist for efficiently factorising arbitrary polynomials into a product of polynomials satisfying these conditions, so that the Cantor–Zassenhaus algorithm can be used to factorise arbitrary polynomials). It gives as output a polynomial <math>g(x)</math> with coefficients in the same field such that <math>g(x)</math> divides <math>f(x)</math>. The algorithm may then be applied recursively to these and subsequent divisors, until we find the decomposition of <math>f(x)</math> into powers of irreducible polynomials (recalling that the [[ring (mathematics)\|ring]] of polynomials over any field is a [[unique factorisation domain]]).

			All possible factors of <math>f(x)</math> are contained within the [[factor ring]]
			<math>R = \frac{\mathbb{F}_q[x]}{\langle f(x) \rangle}</math>. If we suppose that <math>f(x)</math> has irreducible factors <math>p_1(x), p_2(x), \ldots, p_s(x)</math>, all of degree ''d'', then this factor ring is isomorphic to the [[direct product]] of factor rings <math>S = \prod_{i=1}^s \frac{\mathbb{F}_q[x]}{\langle p_i(x) \rangle}</math>. The isomorphism from ''R'' to ''S'', say <math>\phi</math>, maps a polynomial <math>g(x) \in R</math> to the ''s''-tuple of its reductions modulo each of the <math>p_i(x)</math>, i.e. if:

			:<math>
			\begin{align}
			g(x) & {} \equiv g_1(x) \pmod{p_1(x)}, \\
			g(x) & {} \equiv g_2(x) \pmod{p_2(x)}, \\
			& {} \ \ \vdots \\
			g(x) & {} \equiv g_s(x) \pmod{p_s(x)},
			\end{align}
			</math>

			then <math>\phi(g(x) + \langle f(x) \rangle) = (g_1(x) + \langle p_1(x) \rangle, \ldots, g_s(x) + \langle p_s(x) \rangle)</math>. It is important to note the following at this point, as it shall be of critical importance later in the algorithm: Since the <math>p_i(x)</math> are each irreducible, each of the factor rings in this direct sum is in fact a field. These fields each have degree <math>q^d</math>.

			===Core result===
			The core result underlying the Cantor–Zassenhaus algorithm is the following: If <math>a(x) \in R</math> is a polynomial satisfying:

			: <math>a(x) \neq 0, \pm 1 </math>

			: <math>a_i(x) \in \{0,-1,1\}\text{ for }i=1,2,\ldots, s,</math>

			where <math>a_i(x)</math> is the reduction of <math>a(x)</math> modulo <math>p_i(x)</math> as before, and if any two of the following three sets is non-empty:

			: <math>A = \{ i \| a_i(x) = 0 \}, </math>

			: <math>B = \{ i \| a_i(x) = -1 \}, </math>

			: <math>C = \{ i \| a_i(x) = 1 \}, </math>

			then there exist the following non-trivial factors of <math>f(x)</math>:

			: <math>\gcd(f(x),a(x)) = \prod_{i \in A} p_i(x),</math>

			: <math>\gcd(f(x),a(x)+1) = \prod_{i \in B} p_i(x),</math>

			: <math>\gcd(f(x),a(x)-1) = \prod_{i \in C} p_i(x).</math>

			===Algorithm===
			The Cantor–Zassenhaus algorithm computes polynomials of the same type as <math>a(x)</math> above using the isomorphism discussed in the Background section. It proceeds as follows, in the case where the field <math>\mathbb{F}_q</math> is of odd-characteristic. The process can be generalised to characteristic 2 fields in a fairly straightforward way: Select a random polynomial <math>b(x) \in R</math> such that <math>b(x) \neq 0, \pm 1 </math>. Set <math>m=(q^d-1)/2</math> and compute <math>b(x)^m</math>. Since <math>\phi</math> is an isomorphism, we have (using our now-established notation):

			:<math>\phi(b(x)^m) = (b_1^m(x) + \langle p_1(x) \rangle, \ldots, b^m_s(x) + \langle p_s(x) \rangle).</math>

			Now, each <math>b_i(x) + \langle p_i(x)\rangle</math> is an element of a field of order <math>q^d</math>, as noted earlier. The multiplicative subgroup of this field has order <math>q^d-1</math> and so, unless <math>b_i(x)=0</math>, we have <math>b_i(x)^{q^d-1}=1</math> for each ''i'' and hence <math>b_i(x)^m = \pm 1</math> for each ''i''. If <math>b_i(x)=0</math>, then of course <math>b_i(x)^m=0</math>. Hence <math>b(x)^m</math> is a polynomial of the same type as <math>a(x)</math> above. Further, since <math>b(x) \neq 0, \pm1</math>, at least two of the sets <math>A,B</math> and ''C'' are non-empty and by computing the above GCDs we may obtain non-trivial factors. Since the ring of polynomials over a field is an [[Euclidean domain]], we may compute these GCDs using the [[Euclidean algorithm]].

			==Applications==
			One important application of the Cantor–Zassenhaus algorithm is in computing [[discrete logarithm]]s over finite fields of prime-power order. Computing discrete logarithms is an important problem in [[public key cryptography]]. For a field of prime-power order, the fastest known method is the [[index calculus\|index calculus method]], which involves the factorisation of field elements. If we represent the prime-power order field in the usual way – that is, as polynomials over the prime order base field, reduced modulo an irreducible polynomial of appropriate degree – then this is simply polynomial factorisation, as provided by the Cantor–Zassenhaus algorithm.

			==Implementation in computer algebra systems==
			The Cantor–Zassenhaus algorithm may be accessed in the PARI/GP package using the [http://pari.math.u-bordeaux.fr/dochtml/html.stable/Arithmetic_functions.html#factorcantor factorcantor] command.

			==See also==
			*[[Polynomial factorization\|Polynomial factorisation]]
			*[[Factorization of polynomials over a finite field and irreducibility tests]]
			*[[Berlekamp's algorithm]]

			==References==
			* {{cite journal\|title=A New Algorithm for Factoring Polynomials Over Finite Fields
			\|author=David G. Cantor, Hans Zassenhaus
			\|journal=Mathematics of Computation
			\|volume=36
			\|issue=154
			\|date=April 1981
			\|pages=587–592
			\|jstor=2007663
			\|mr=606517
			\|ref=harv
			\|doi=10.1090/S0025-5718-1981-0606517-5 }}

			{{DEFAULTSORT:Cantor-Zassenhaus algorithm}}
			[[Category:Computer algebra]]
			[[Category:Finite fields]]

en>Rcampbell1: /* References */

2012-05-09T12:53:25Z

References

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Hello. Allow me introduce the writer. Her name is Emilia Shroyer but it's not the most feminine name out there. To collect coins is 1 of the things I adore most. California is our birth location. He used to be unemployed but now he is a computer operator but his marketing never comes.<br><br>Visit my web-site - at home std testing ([http://test.ithink-now.org/content/think-f83d96d08fe71e25bf80a6cb59048a18 browse around this website])

Berlekamp's algorithm - Revision history

en>Qwertyus: I don't see why PARI/GP deserves special mention, in the lede

en>Xqbot: Robot: ru:Алгоритм Берлекэмпа is a good article

en>Rcampbell1: /* References */