The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Solvability by radicals is in polynomial time
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Riemann hypothesis and finding roots over finite fields
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Solvability by radicals from an algorithmic point of view
Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Factoring polynomials over finite fields
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Algorithms for solving linear systems over cyclotomic fields
Journal of Symbolic Computation
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Based on Kummer Theorem, we study the deterministic complexity of two factorization problems: polynomial factorization over finite fields and prime factorization in algebraic number fields. We show that factoring polynomials of degree n in Fp[x], with p prime, is polynimially equivalent to factoring p in algebraic number field of extension degree n over Q, where p is “regular” with respect to the generating polynomials of the number fields. Part of the proof also yields an efficient polynomial time algorithm for computing the factorization pattern. Number theoretical methods are then developed to solve two important kinds of polynomials:&fgr;n(x) mod p where &fgr;n is the n-th cyclotomic polynomial, and xn. - &agr; mod p where &agr; &egr; N. We show that when Extended Riemann Hypothesis is assumed, all the roots of both kinds of polynomials in Fp can be found efficiently in time polynomial in n and logp. As &agr; consequence, when p &Xgr; 1(n), factorization of p in the n-th cyclotomic field can be computed in polynomial time. The result on finding all roots of xn &Xgr; &agr;(p) extends &agr; result of Adleman, Menders, and Miller, which states that the least root of xn &Xgr; &agr;(p) can be found in polynomial time, when Extended Riemann Hypothesis is assumed.