Motivation Factorization of cyclotomic polynomials

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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

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English

Mestre Algorithm

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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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IThere are deterministic algorithms inFp[X](egBerlekamp’s algorithm) but exponential in logp. INo deterministic polynomial-time algorithm is known for factorization inFp[X] in degree 2 !. Even IEasy to decide ifa∈Fpis a square (Legendre symbol, or more generally the g.c.d. withxp−x) IA lot of literature for square root probabilistic-algorithms, but as for now, we don’t know if it’s aP–problem. IHowever, thanks toSchoof’s algorithm, we can say some-thing indeterministictime.

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IThere are deterministic algorithms inFp[X](egBerlekamp’s algorithm) but exponential in logp. INo deterministic polynomial-time algorithm is known for factorization inFp[X]. Even in degree 2 ! IEasy to decide ifa∈Fpis a square (Legendre symbol, or more generally the g.c.d. withxp−x) IA lot of literature for square root probabilistic-algorithms, but as for now, we don’t know if it’s aP–problem. IHowever, thanks toSchoof’s algorithm, we can say some-thing indeterministictime.

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IThere are deterministic algorithms inFp[X](egBerlekamp’s algorithm) but exponential in logp. INo deterministic polynomial-time algorithm is known for factorization inFp[X]. Even in degree 2 ! IEasy to decide ifa∈Fpis a square (Legendre symbol, or more generally the g.c.d. withxp−x) IA lot of literature for square root probabilistic-algorithms, but as for now, we don’t know if it’s aP–problem. IHowever, thanks toSchoof’s algorithm, we can say some-thing indeterministictime.