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Elliptic Curves And Primality Proving
 Math. Comp
, 1993
"... The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm. ..."
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Cited by 202 (22 self)
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The aim of this paper is to describe the theory and implementation of the Elliptic Curve Primality Proving algorithm.
Searching for Primitive Roots in Finite Fields
, 1992
"... Let GF(p n ) be the finite field with p n elements where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(p n ) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(p n ). ..."
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Cited by 51 (3 self)
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Let GF(p n ) be the finite field with p n elements where p is prime. We consider the problem of how to deterministically generate in polynomial time a subset of GF(p n ) that contains a primitive root, i.e., an element that generates the multiplicative group of nonzero elements in GF(p n ). We present three results. First, we present a solution to this problem for the case where p is small, i.e., p = n O(1) . Second, we present a solution to this problem under the assumption of the Extended Riemann Hypothesis (ERH) for the case where p is large and n = 2. Third, we give a quantitative improvement of a theorem of Wang on the least primitive root for GF(p) assuming the ERH. Appeared in Mathematics of Computation 58, pp. 369380, 1992. An earlier version of this paper appeared in the 22nd Annual ACM Symposium on Theory of Computing (1990), pp. 546554. 1980 Mathematics Subject Classification (1985 revision): 11T06. 1. Introduction Consider the problem of finding a primitive ...
Factorization of Polynomials Given by StraightLine Programs
 Randomness and Computation
, 1989
"... An algorithm is developed for the factorization of a multivariate polynomial represented by traightline program into its irreducible factors. The algorithm is in random polynomialtime as a function in the input size, total degree, and binary coefficient length for the usual coefficient fields and ..."
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Cited by 36 (8 self)
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An algorithm is developed for the factorization of a multivariate polynomial represented by traightline program into its irreducible factors. The algorithm is in random polynomialtime as a function in the input size, total degree, and binary coefficient length for the usual coefficient fields and outputs a straightline program, which with controllably high probability correctly determines the irreducible factors. It also returns the probably correct multiplicities of each distinct factor. If th oefficient field has finite characteristic p and p divides the multiplicities of some irreducible factors our algorithm constructs straightline programs for the appropriate pth powers of such factors. Also a probabilistic algorithm is presented that allows to convert a polynomial given by a straightline program into its sparse representation. This conversion algorithm is in randompolynomial time in the previously cited parameters and in an upper bound for the number of nonzero...
Constructing nonresidues in finite fields and the extended Riemann hypothesis
 Math. Comp
, 1991
"... Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in pol ..."
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Cited by 12 (0 self)
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Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomialtime bound holds even if k is exponentially large. More generally, assuming the ERH, in time (n log p) O(n) we can construct a set of elements
Deterministic Irreducibility Testing of Polynomials over Large Finite Fields
 J. Symbolic Comput
, 1987
"... We present a sequential deterministic polynomialtime algorithm for testing dense multivariate polynomials over a large finite field for irreducibility. All previously known algorithms were of a probabilistic nature. Our deterministic solution is based on our algorithm for absolute irreducibility te ..."
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Cited by 8 (3 self)
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We present a sequential deterministic polynomialtime algorithm for testing dense multivariate polynomials over a large finite field for irreducibility. All previously known algorithms were of a probabilistic nature. Our deterministic solution is based on our algorithm for absolute irreducibility testing combined with Berlekamp's algorithm.
Construction Of Hilbert Class Fields Of Imaginary Quadratic Fields And Dihedral Equations Modulo p
, 1989
"... . The implementation of the AtkinGoldwasserKilian primality testing algorithm requires the construction of the Hilbert class field of an imaginary quadratic field. We describe the computation of a defining equation for this field in terms of Weber's class invariants. The polynomial we obtain, ..."
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Cited by 5 (3 self)
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. The implementation of the AtkinGoldwasserKilian primality testing algorithm requires the construction of the Hilbert class field of an imaginary quadratic field. We describe the computation of a defining equation for this field in terms of Weber's class invariants. The polynomial we obtain, noted W(X), has a solvable Galois group. When this group is dihedral, we show how to express the roots of this polynomial in terms of radicals. We then use these expressions to solve the equation W(X) j 0 mod p, where p is a prime. 1 Hilbert polynomials 1.1 Weber's functions We first introduce some functions. Let z be any complex number and put q = exp(2ißz). Dedekind's j function is defined by [21, x24 p. 85] j(z) = j(q) = q 1=24 Y m1 (1 \Gamma q m ): (1) We can expand j as [21, x34 p. 112] j(q) = q 1=24 0 @ 1 + X n1 (\Gamma1) n (q n(3n\Gamma1)=2 + q n(3n+1)=2 ) 1 A : (2) The Weber's functions are [21, x34 p. 114] f(z) = e \Gammaiß=24 j( z+1 2 ) j(z) ; (3) f 1 (z) = j...
Cyclotomy primality proofs and their certificates. Mathematica Goettingensis
, 2006
"... Elle est à toi cette chanson Toi l’professeur qui sans façon, As ouvert ma petite thèse Quand mon espoir manquait de braise 1. To the memory of Manuel Bronstein ..."
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Cited by 4 (2 self)
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Elle est à toi cette chanson Toi l’professeur qui sans façon, As ouvert ma petite thèse Quand mon espoir manquait de braise 1. To the memory of Manuel Bronstein
Removing Randomness From Computational Number Theory
, 1989
"... In recent years, many probabilistic algorithms (i.e., algorithms that can toss coins) that run in polynomial time have been discovered for problems with no known deterministic polynomial time algorithms. Perhaps the most famous example is the problem of testing large (say, 100 digit) numbers for pri ..."
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Cited by 3 (1 self)
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In recent years, many probabilistic algorithms (i.e., algorithms that can toss coins) that run in polynomial time have been discovered for problems with no known deterministic polynomial time algorithms. Perhaps the most famous example is the problem of testing large (say, 100 digit) numbers for primality. Even for problems which are known to have deterministic polynomial time algorithms, these algorithms are often not as fast as some probabilistic algorithms for the same problem. Even though probabilistic algorithms are useful in practice, we would like to know, for both theoretical and practical reasons, if randomization is really necessary to obtain the most efficient algorithms for certain problems. That is, we would like to know for which problems there is an inherent gap between the deterministic and probabilistic complexities of these problems. In this research, we consider two problems of a number theoretic nature: factoring polynomials over finite fields and constructing irred...
Trading GRH for Algebra: Algorithms for Factoring Polynomials and Related Structures
, 2009
"... In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a polynomial f(x) of degree n over a finite field k, we can fi ..."
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Cited by 2 (1 self)
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In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a polynomial f(x) of degree n over a finite field k, we can find in deterministic poly(n log n, log k) time either a nontrivial factor of f(x) or a nontrivial automorphism of k[x]/(f(x)) of order n. This main tool leads to various new GRHfree results, most striking of which are: 1. Given a noncommutative algebra A of dimension n over a finite field k. There is a deterministic poly(n log n, log k) time algorithm to find a zero divisor in A. This is the best known deterministic GRHfree result since Friedl and Rónyai (STOC 1985) first studied the problem of finding zero divisors in finite algebras and showed that this problem has the same complexity as factoring polynomials over finite fields. 2. Given a positive integer r such that either 8r or r has at least two distinct odd prime factors. There is a deterministic polynomial time algorithm to find a nontrivial factor of the rth cyclotomic polynomial over a finite field. This is the best known deterministic GRHfree result since Huang (STOC 1985) showed that cyclotomic polynomials can be factored over finite fields in deterministic polynomial time assuming GRH. In this paper, following the seminal work of Lenstra (1991) on constructing isomorphisms between finite fields, we further generalize classical Galois theory constructs like cyclotomic extensions, Kummer extensions, Teichmüller subgroups, to the case of commutative semisimple algebras with automorphisms. These generalized constructs help eliminate the dependence on GRH.
ELLIPTIC GAUSS SUMS AND APPLICATIONS TO POINTCOUNTING.
"... We shall first briefly review some general facts on elliptic curves over finite fields, and the algorithms of Schoof and Schoof Elkies Atkin (SEA) for counting points. We refer to text books as Cox, Silverman or Washington [Cox, Si, Wa] for the general topics and to the original papers of Schoof [ ..."
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We shall first briefly review some general facts on elliptic curves over finite fields, and the algorithms of Schoof and Schoof Elkies Atkin (SEA) for counting points. We refer to text books as Cox, Silverman or Washington [Cox, Si, Wa] for the general topics and to the original papers of Schoof [Sch, Sch1] for a presentation of the algorithms.