Motivation Factorization of cyclotomic polynomials

Ivan Boyer

Description

Motivation Factorization of cyclotomic polynomials Some steps to abelian extensions Conclusion Factorization of polynomials over finite fields in deterministic polynomial-time Ivan Boyer Doctorant sous la direction de Jean-François Mestre Institut Mathématique de Jussieu AGCT-13 — C.I.R.M. March 15, 2011 I. Boyer Factorization in Fp[X]

• context schoof's


• algorithm


• root probabilistic-algorithms


• deterministic polynomial


• motivation factorization


• doctorant sous la direction


• polynomials over

Sujets

• Mestre
• Algorithm

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 24
• Langue: English

Extrait

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Factorization of polynomials over finite fields in deterministicopylimenomial-t

Doctorant sous la direction de Jean-François Mestre Institut Mathématique de Jussieu

AGCT-13 — C.I.R.M. March 15, 2011

Ivan Boyer

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Remark Thedeterministicaspect iscrucial be- everythingin this talk : comes “trivial” in probabilistic time. In the same way, assuming G.R.H. would withdraw some of the interest of the following !

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I There are deterministic algorithms inF p [X](egBerlekamp’s algorithm) but exponential in logp. I No deterministic polynomial-time algorithm is known for factorization inF p [X] in degree 2 !. Even I Easy to decide ifa∈F p is a square (Legendre symbol, or more generally the g.c.d. withx p −x) I A lot of literature for square root probabilistic-algorithms, but as for now, we don’t know if it’s aP–problem. I However, thanks toSchoof’s algorithm, we can say some- thing indeterministictime.

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I There are deterministic algorithms inF p [X](egBerlekamp’s algorithm) but exponential in logp. I No deterministic polynomial-time algorithm is known for factorization inF p [X]. Even in degree 2 ! I Easy to decide ifa∈F p is a square (Legendre symbol, or more generally the g.c.d. withx p −x) I A lot of literature for square root probabilistic-algorithms, but as for now, we don’t know if it’s aP–problem. I However, thanks toSchoof’s algorithm, we can say some- thing indeterministictime.

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I There are deterministic algorithms inF p [X](egBerlekamp’s algorithm) but exponential in logp. I No deterministic polynomial-time algorithm is known for factorization inF p [X] in degree 2 !. Even I Easy to decide ifa∈F p is a square (Legendre symbol, or more generally the g.c.d. withx p −x) I A lot of literature for square root probabilistic-algorithms, but as for now, we don’t know if it’s aP–problem. I However, thanks toSchoof’s algorithm, we can say some- thing indeterministictime.

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I There are deterministic algorithms inF p [X](egBerlekamp’s algorithm) but exponential in logp. I No deterministic polynomial-time algorithm is known for factorization inF p [X]. Even in degree 2 ! I Easy to decide ifa∈F p is a square (Legendre symbol, or more generally the g.c.d. withx p −x) I A lot of literature for square root probabilistic-algorithms, but as for now, we don’t know if it’s aP–problem. I However, thanks toSchoof’s algorithm, we can say some- thing indeterministictime.

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