Factoring cyclotomic polynomials over large finite fields

Stein, Greg

doi:10.1017/cbo9780511525988.026

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Greg Stein1

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2012

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Cited by 3 publications

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“…In this section we prove a result first presented in [20]. The original work did not contain the explicit description of the algorithms used or as sharp a running time analysis.…”

Section: Greg Steinmentioning

confidence: 80%

Using the theory of cyclotomy to factor cyclotomic polynomials over finite fields

Stein

2000

Math. Comp.

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Abstract. We examine the problem of factoring the rth cyclotomic polynomial, Φr(x), over Fp, r and p distinct primes. Given the traces of the roots of Φr(x) we construct the coefficients of Φr(x) in time O(r 4 ). We demonstrate a deterministic algorithm for factoring Φr(x) in time O((r 1/2+ log p) 9 ) when Φr(x) has precisely two irreducible factors. Finally, we present a deterministic algorithm for computing the sum of the irreducible factors of Φr(x) in time O(r 6 ).

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“…In this section we prove a result first presented in [20]. The original work did not contain the explicit description of the algorithms used or as sharp a running time analysis.…”

Section: Greg Steinmentioning

confidence: 80%

Using the theory of cyclotomy to factor cyclotomic polynomials over finite fields

Stein

2000

Math. Comp.

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An Alternate Construction of the Berlekamp Subalgebra

Stein

2002

Finite Fields With Applications to Coding Theory, Cryptography and Related Areas

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Trading GRH for algebra: Algorithms for factoring polynomials and related structures

et al. 2012

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Abstract. In this paper we develop a general technique to eliminate the assumption of the Generalized Riemann Hypothesis (GRH) from various deterministic polynomial factoring algorithms over finite fields. It is the first bona fide progress on that issue for more than 25 years of study of the problem. Our main results are basically of the following form: either we construct a nontrivial factor of a given polynomial or compute a nontrivial automorphism of the factor algebra of the given polynomial. Probably the most notable application of such automorphisms is efficiently finding zero divisors in noncommutative algebras. The proof methods used in this paper exploit virtual roots of unity and lead to efficient actual polynomial factoring algorithms in special cases.

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Factoring cyclotomic polynomials over large finite fields

Cited by 3 publications

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Using the theory of cyclotomy to factor cyclotomic polynomials over finite fields

Using the theory of cyclotomy to factor cyclotomic polynomials over finite fields

An Alternate Construction of the Berlekamp Subalgebra

Trading GRH for algebra: Algorithms for factoring polynomials and related structures

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