Factorization of solvable polynomials over finite fields and the generalized Riemann hypothesis
- 44 Downloads
This article presents an algorithm that, assuming the generalized Riemann hypothesis, factors a polynomial f mod p, where f ∃Z[X] is solvable overQ, into irreducible (over the fieldFpm) factors in time polynomial in m, log p, and the length of notation of f. The following problems are also solved in time polynomial in m, n, and log p: 1) construction of the fieldFpm, 2) construction of all isomorphisms between two realizations ofFpm, and 3) computation of the roots of degree n inFpm.
Unable to display preview. Download preview PDF.
- 1.M.-D. A. Huang, “Riemann hypothesis and finding roots over finite fields,” in: Proc. Seventeenth ACM Symp. on Theory of Computing, New York (1985), pp. 121–130.Google Scholar
- 2.S. Landau and G. Miller, “Solvability by radicals is in polynomial time,” in: Proc. Fifteenth ACM Symp. on Theory of Computing, New York (1983), pp. 140–151.Google Scholar
- 3.S. Lang, Algebra, Addison-Wesley, Reading (1965).Google Scholar
- 4.S. Lang, Algebraic Number Theory, Addison-Wesley, Reading (1970).Google Scholar
- 5.Z. I. Borevich and I. R. Shafarevich, Number Theory [in Russian], Moscow (1964).Google Scholar
- 6.J. C. Lagarias, H. L. Montgomery, and A. M. Odlyzko, “A bound for the least prime ideal in the Chebotarev density theorem,” Inv. Math.,54, 271–296 (1979).Google Scholar
- 7.I. M. Vinogradov, Fundamentals of Number Theory [in Russian], Moscow (1972).Google Scholar
- 8.E. R. Berlekamp, “Factoring polynomials over finite fields,” Bell Sysem Tech. J.,48, 1853–1859 (1967).Google Scholar
- 9.A. K. Lenstra, H. W. Lenstra, and L. Lovasz, “Factoring polynomials with rational coefficients,” Math. Ann.,261, 515–535 (1982).Google Scholar
- 10.A. L. Chistov, “Polynomial complexity algorithm for factoring polynomials and finding components of varieties in subexponential time,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Akad. Nauk SSSR,137, 124–188 (1984).Google Scholar
- 11.L. Ronyai, “Factoring polynomials over finite fields,” in: Proc. Twenty Eighth IEEE Found. Comput. Sci., New York (1987).Google Scholar
- 12.S. A. Evdokimov, “The generalized Riemann hypothesis and factorization of solvable polynomials over finite fields,” in: Abstracts of the Eighth All-Union Conf. on Math. Logic [in Russian] (1986), p. 64.Google Scholar