Journal Article10.1080/10586458.1999.10504623
Sieving in function fields
Ralf Flassenberg,Sachar Paulus +1 more
TL;DR: The first implementation of sieving techniques in the context of function fields is presented, combining the algorithm of Hafner and McCurley with sieving ideas known from factoring to compute in class groups of quadratic congruence function fields.
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Abstract: We present the first implementation of sieving techniques in the context of function fields. More precisely, we compute in class groups of quadratic congruence function fields by combining the algorithm of Hafner and McCurley with sieving ideas known from factoring. We apply our methods to the computation of generators and relations of the Jacobian variety of hyperelliptic curves over fin ite fields. The algorithms introduced here were implemented in C++ with the help of LEDA and LiDIA. We provide examples of running times and comparisons with earlier algorithms.
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Citations
Computing discrete logarithms in the Jacobian of high-genus hyperelliptic curves over even characteristic finite fields
TL;DR: In this paper, improved versions of index-calculus algorithms for solving discrete logarithm problems in Jacobians of high-genus hyperelliptic curves are described.
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A Simple Method for Obtaining Relations Among Factor Basis Elements for Special Hyperelliptic Curves.
Palash Sarkar,Shashank Singh +1 more
TL;DR: In this article, the authors revisited Nagao's approach with the idea of avoiding the requirement of solving a multi-variate system of non-linear equations and showed that it is possible to obtain a single relation in about (2g + 3)! trials.
•Posted Content
Discrete logarithms in curves over finite fields
TL;DR: A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields and their applications in discrete-log- LaSalle inequality.
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A New Method for Decomposition in the Jacobian of Small Genus Hyperelliptic Curves.
Palash Sarkar,Shashank Singh +1 more
TL;DR: A new method for decomposition over the same factor basis in the Jacobian of a hyperelliptic curve is provided, which admits a sieving technique which removes smoothness checking of polynomials required in Gaudry’s method.
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Computing in the Jacobian of a hyperelliptic curve
TL;DR: A reduction algorithm is presented which is asymptotically faster than that of Gauss when the genus g is very large and the Jacobian of a hyperelliptic curve is studied.
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Algorithms in number theory
Arjen K. Lenstra,Hendrik W. Lenstra +1 more
- 02 Jan 1991
TL;DR: This chapter discusses algorithms that solve two basic problems in computational number theory—factoring integers into prime factors and finding discrete logarithms.
Algorithms in number theory
Arjen K. Lenstra,Hendrik W. Lenstra +1 more
- 01 Jan 1990
TL;DR: In this article, the authors discuss algorithms that solve two basic problems in computational number theory (factoring integers into prime factors and finding discrete logarithm) and their analyses depend on many different parts of number theory.
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