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Journal ArticleDOI

Generalized Riemann hypothesis and factoring polynomials over finite fields

01 Sep 1991-Journal of Algorithms (Academic Press, Inc.)-Vol. 12, Iss: 3, pp 464-481
TL;DR: It is shown that, assuming the generalized Riemann hypothesis, there exists a deterministic polynomial time algorithm, which on input of a rational prime p and a monic integral polynometric f computes all the irreducible factors of f mod p in F p.
About: This article is published in Journal of Algorithms.The article was published on 1991-09-01. It has received 28 citations till now. The article focuses on the topics: Minimal polynomial (linear algebra) & Monic polynomial.
Citations
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Book ChapterDOI
06 Apr 1992
TL;DR: This article discusses important developments of the past five years in factorization and discusses the “classical univariate problems” of factoring a polynomial.
Abstract: Algorithms invented in the past 25 years make it possible on a computer to efficiently factor a polynomial in one, several, or many variables with coefficients from a certain field, such as a finite field or the rational, real, or complex numbers. I have surveyed work up to 1986 in the papers (Kaltofen 1982 and 1990a). This article discusses important developments of the past five years; I also take a fresh perspective of some older results. Although a conscientious effort has been made to cover (at least by citation) the significant contributions of that period, omissions are likely, which I ask to be kindly brought to my attention. Three parameters partition the factorization problem: first, the mathematical nature and computational representation of the coefficient domains of the input polynomial, second, that of the irreducible factors, and, third, the representation of the input polynomial and the sought irreducible factors, which depends not only on the degree and number of variables but also on properties such as sparsity. Say, for instance, that a bivariate polynomial with rational coefficients is to be factored into irreducible polynomials with real coefficients. The input polynomial as well as the factors may be represented by lists of monomials, that is terms and their corresponding non-zero coefficients. For the rational input the coefficients can be just fractions of two long integers, but the representation of the real coefficients for the factors is less standardized. One choice represents a real algebraic number by its rational minimum polynomial and an isolating interval with rational boundaries (Collins 1975), while another uses a rational linear relation of powers of a complex algebraic number that is universal for all coefficients of a single factor (Kaltofen 1990b). The organization of this survey is governed by these distinguishing problem specifications. We first discuss the “classical univariate problems” of factoring a polynomial

117 citations

Journal ArticleDOI
TL;DR: This survey reviews several algorithms for the factorization of univariate polynomials over finite fields and emphasizes the main ideas of the methods and provides an up-to-date bibliography of the problem.

115 citations

Book ChapterDOI
06 May 1994
TL;DR: This chapter presents a collection of 36 open problems in number theoretic complexity, showing how questions about the integers have natural generalizations to rings of integers in an algebraic number field, and questions about elliptic curves may generalize to arbitrary abelian varieties.
Abstract: Publisher Summary In the past decade, there has been a resurgence of interest in computational problems of a number theoretic nature. This period has been characterized by a growing awareness of the practical aspects of number theoretic computations and at the same time by an increased understanding of the relevance of deep theory to the problems that arise. This chapter presents a collection of 36 open problems in number theoretic complexity. Questions about the integers have natural generalizations to rings of integers in an algebraic number field, and questions about elliptic curves may generalize to arbitrary abelian varieties. The chapter presents the problems that arose from many different places and times.

73 citations

Journal ArticleDOI
Shuhong Gao1
TL;DR: It is proved that a proper factor of a polynomials can be found deterministically in polynomial time, under ERH, if its roots do not satisfy some stringent condition, called super square balanced.

28 citations

Journal ArticleDOI
TL;DR: New deterministic algorithms, based on Graeffe transforms, to compute all the roots of a polynomial which splits over a finite field, and a new nearly optimal algorithm for computing characteristic polynomials of multiplication endomorphisms in finite field extensions.
Abstract: We design new deterministic algorithms, based on Graeffe transforms, to compute all the roots of a polynomial which splits over a finite field $$\mathbb {F}_q$$Fq. Our algorithms were designed to be particularly efficient in the case when the cardinality $$q - 1$$q-1 of the multiplicative group of $$\mathbb {F}_q$$Fq is smooth. Such fields are often used in practice because they support fast discrete Fourier transforms. We also present a new nearly optimal algorithm for computing characteristic polynomials of multiplication endomorphisms in finite field extensions. This algorithm allows for the efficient computation of Graeffe transforms of arbitrary orders.

24 citations


Cites methods from "Generalized Riemann hypothesis and ..."

  • ...Schoof, Rónyai, Huang, and Źrałek designed different methods for particular types of input polynomials according to their syntax or to properties of the Galois group of the lifted input polynomial overQ [21,36,37,39,50]....

    [...]

References
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Book
01 Jan 1974
TL;DR: This text introduces the basic data structures and programming techniques often used in efficient algorithms, and covers use of lists, push-down stacks, queues, trees, and graphs.
Abstract: From the Publisher: With this text, you gain an understanding of the fundamental concepts of algorithms, the very heart of computer science. It introduces the basic data structures and programming techniques often used in efficient algorithms. Covers use of lists, push-down stacks, queues, trees, and graphs. Later chapters go into sorting, searching and graphing algorithms, the string-matching algorithms, and the Schonhage-Strassen integer-multiplication algorithm. Provides numerous graded exercises at the end of each chapter. 0201000296B04062001

9,262 citations

Journal ArticleDOI
TL;DR: This paper presents a polynomial-time algorithm to solve the following problem: given a non-zeroPolynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q (X).
Abstract: In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

3,513 citations

01 Jan 1982
TL;DR: In this paper, a polynomial-time algorithm was proposed to decompose a primitive polynomials into irreducible factors in Z(X) if the greatest common divisor of its coefficients is 1.
Abstract: In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

3,248 citations

Book
01 Jan 2015
TL;DR: This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory," originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field.
Abstract: This is the revised edition of Berlekamp's famous book, "Algebraic Coding Theory," originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field. One of these is an algorithm for decoding Reed-Solomon and Bose–Chaudhuri–Hocquenghem codes that subsequently became known as the Berlekamp–Massey Algorithm. Another is the Berlekamp algorithm for factoring polynomials over finite fields, whose later extensions and embellishments became widely used in symbolic manipulation systems. Other novel algorithms improved the basic methods for doing various arithmetic operations in finite fields of characteristic two. Other major research contributions in this book included a new class of Lee metric codes, and precise asymptotic results on the number of information symbols in long binary BCH codes.Selected chapters of the book became a standard graduate textbook.Both practicing engineers and scholars will find this book to be of great value.

2,912 citations

Book
01 Jan 1971
TL;DR: The second edition of Lang's well-known textbook as mentioned in this paper contains a version of a Riemann-Roch theorem in number fields, proved by Lang in the very first version of the book in the sixties.
Abstract: This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. Part I introduces some of the basic ideas of the theory: number fields, ideal classes, ideles and adeles, and zeta functions. It also contains a version of a Riemann-Roch theorem in number fields, proved by Lang in the very first version of the book in the sixties. This version can now be seen as a precursor of Arakelov theory. Part II covers class field theory, and Part III is devoted to analytic methods, including an exposition of Tate's thesis, the Brauer-Siegel theorem, and Weil's explicit formulas. The second edition contains corrections, as well as several additions to the previous edition, and the last chapter on explicit formulas has been rewritten.

2,190 citations