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19
Computing RiemannRoch spaces in algebraic function fields and related topics
, 2001
"... this paper we develop a simple and efficient algorithm for the computation of RiemannRoch spaces to be counted among the arithmetic methods. The algorithm completely avoids series expansions and resulting complications, and instead relies on integral closures and their ideals only. It works for any ..."
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this paper we develop a simple and efficient algorithm for the computation of RiemannRoch spaces to be counted among the arithmetic methods. The algorithm completely avoids series expansions and resulting complications, and instead relies on integral closures and their ideals only. It works for any "computable" constant field k of any characteristic as long as the required integral closures can be computed, and does not involve constant field extensions
A relative van Hoeij algorithm over number fields
 J. Symbolic Computation
, 2004
"... Abstract. Van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as BerlekampZassenhaus, but uses lattice basis reduction to improve drastically on the recombination phase. His ideas give rise to a collection of algorithms, differing greatl ..."
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Abstract. Van Hoeij’s algorithm for factoring univariate polynomials over the rational integers rests on the same principle as BerlekampZassenhaus, but uses lattice basis reduction to improve drastically on the recombination phase. His ideas give rise to a collection of algorithms, differing greatly in their efficiency. We present two deterministic variants, one of which achieves excellent overall performance. We then generalize these ideas to factor polynomials over
A Montgomerylike Square Root for the Number Field Sieve
, 1998
"... The Number Field Sieve (NFS) is the asymptotically fastest factoring algorithm known. It had spectacular successes in factoring numbers of a special form. Then the method was adapted for general numbers, and recently applied to the RSA130 number [6], setting a new world record in factorization. Th ..."
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The Number Field Sieve (NFS) is the asymptotically fastest factoring algorithm known. It had spectacular successes in factoring numbers of a special form. Then the method was adapted for general numbers, and recently applied to the RSA130 number [6], setting a new world record in factorization. The NFS has undergone several modifications since its appearance. One of these modifications concerns the last stage: the computation of the square root of a huge algebraic number given as a product of hundreds of thousands of small ones. This problem was not satisfactorily solved until the appearance of an algorithm by Peter Montgomery. Unfortunately, Montgomery only published a preliminary version of his algorithm [15], while a description of his own implementation can be found in [7]. In this paper, we present a variant of the algorithm, compare it with the original algorithm, and discuss its complexity.
Group structures of elementary supersingular abelian varieties over finite fields
 J. Number Theory
, 2000
"... Let A be a supersingular abelian variety over a finite field k which is kisogenous to a power of a simple abelian variety over k. Write the characteristic polynomial of the Frobenius endomorphism of A relative to k as f = g e for a monic irreducible polynomial g and a positive integer e. We show th ..."
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Cited by 7 (1 self)
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Let A be a supersingular abelian variety over a finite field k which is kisogenous to a power of a simple abelian variety over k. Write the characteristic polynomial of the Frobenius endomorphism of A relative to k as f = g e for a monic irreducible polynomial g and a positive integer e. We show that the group of krational points A(k) onAis isomorphic to (Z g(1) Z) e unless A's simple component is of dimension 1 or 2, in which case we prove that A(k) is isomorphic to (Z g(1) Z) a _ (Z (g(1) 2) Z_Z 2Z) b for some nonnegative integers a, b with a+b=e. In particular, if the characteristic of k is 2 or A is simple of dimension greater than 2, then A(k)$(Z g(1) Z) e.
Identifying the Matrix Ring: ALGORITHMS FOR QUATERNION ALGEBRAS AND QUADRATIC FORMS
, 2010
"... We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2matrix ring M2(R) and, if so, to compute such an embedding. We d ..."
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We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 × 2matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders.
An explicit treatment of biquadratic function fields
 Contributions to Discrete Mathematics
"... Abstract. We provide a comprehensive description of biquadratic function fields and their properties, including a characterization of the cyclic and radical cases as well as the constant field. For the cyclic scenario, we provide simple explicit formulas for the ramification index of any rational pl ..."
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Abstract. We provide a comprehensive description of biquadratic function fields and their properties, including a characterization of the cyclic and radical cases as well as the constant field. For the cyclic scenario, we provide simple explicit formulas for the ramification index of any rational place, the field discriminant, the genus, and an algorithmically suitable integral basis. In terms of computation, we only require square and fourth power testing of constants, extended gcd computations of polynomials, and the squarefree factorization of polynomials over the base field. 1.
A polynomial with Galois group SL2(F16)
"... In this paper we display an explicit polynomial having Galois group SL2(F16), filling in a gap in the tables of Jürgen Klüners and Gunter Malle. Furthermore, the polynomial has small Galois root discriminant; this fact answers a question of John Jones and David Roberts. The computation of this polyn ..."
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In this paper we display an explicit polynomial having Galois group SL2(F16), filling in a gap in the tables of Jürgen Klüners and Gunter Malle. Furthermore, the polynomial has small Galois root discriminant; this fact answers a question of John Jones and David Roberts. The computation of this polynomial uses modular forms and their Galois representations. This paper has been published in the LCM Journal of Computation and Mathematics, volume 10 (2007), pages 378–388. 1