Factoring Polynomials over Special Finite Fields

Abstract

DEDICATED TO CHAO KO FOR HIS 90TH BIRTHDAYWe exhibit a deterministic algorithm for factoring polynomials in one variable over "nite "elds. It is e$cient only if a positive integer k is known for which I (p) is built up from small prime factors; here I denotes the kth cyclotomic polynomial, and p is the characteristic of the "eld. In the case k"1, when I (p)"p!1, such an algorithm was known, and its analysis required the generalized Riemann hypothesis. Our algorithm depends on a similar, but weaker, assumption; speci"cally, the algorithm requires the availability of an irreducible polynomial of degree r over Z/pZ for each prime number r for which I (p) has a prime factor l with l,1 mod r. An auxiliary procedure is devoted to the construction of roots of unity by means of Gauss sums. We do not claim that our algorithm has any practical value.2000 Academic Press

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

“…The most notable exception which withstood our efforts is the result of Bach, von zur Gathen and Lenstra [BGL01] for the case when Φ k (p) is smooth. An even more important limitation of our results is that they do not provide (direct) tools for computing square, cubic, etc., roots in general finite fields.…”

Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field <I>GF</I>(<I>p<sup>q</sup></I>)

“…After that a new variant of Rabin's algorithm [27] had been introduced with probabilistic analysis of BPs with no irreducible factors [28]. Later a factorization of univariate Polynomials Over Galois Field GF(p) in sub quadratic execution time had also been notified [29].…”

On the Deterministic Complexity of Factoring Polynomials

“…Suppose that q − 1 = r e w where r w. Let η be a fixed primitive r e th root of unity in F q . We remark that η can be taken as ξ w for any primitive root or rth nonresidue ξ in F q and ξ can be constructed efficiently assuming ERH (Wang, 1959;Bach, 1997).…”

“…If the number of irreducible factors of a polynomial is bounded, Rónyai (1988) showed under ERH that it can be factored deterministically in polynomial time. On special fields, Bach et al (1995) showed that polynomials over finite fields of characteristic p can be factored in polynomial time if Φ k (p) is smooth for some integer k where Φ k (x) denotes the kth cyclotomic polynomial, which extends the works of von zur Gathen (1987); Moenck (1977); Camion (1983); Mignotte and Schnorr (1988), and Rónyai (1989). Recently, Evdokimov (1994) proved that every polynomial over F q of degree n can be factored deterministically in time polynomial in n log n and log q.…”