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Improvements in the computation of ideal class groups of imaginary quadratic number fields
 ADVANCES IN MATHEMATICS OF COMPUTATION
"... We investigate improvements to the algorithm for the computation of ideal class group described by Jacobson in the imaginary quadratic case. These improvements rely on the large prime strategy and a new method for performing the linear algebra phase. We achieve a significant speedup and are able ..."
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We investigate improvements to the algorithm for the computation of ideal class group described by Jacobson in the imaginary quadratic case. These improvements rely on the large prime strategy and a new method for performing the linear algebra phase. We achieve a significant speedup and are able to compute 110decimal digits discriminant ideal class group in less than a week.
COMPUTING DISCRETE LOGARITHMS IN THE JACOBIAN OF HIGHGENUS HYPERELLIPTIC CURVES OVER EVEN CHARACTERISTIC FINITE FIELDS
"... Abstract. We describe improved versions of indexcalculus algorithms for solving discrete logarithm problems in Jacobians of highgenus hyperelliptic curves de ned over even characteristic elds. Our rst improvement is to incorporate several ideas for the lowgenus case by Gaudry and Theriault, inclu ..."
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Abstract. We describe improved versions of indexcalculus algorithms for solving discrete logarithm problems in Jacobians of highgenus hyperelliptic curves de ned over even characteristic elds. Our rst improvement is to incorporate several ideas for the lowgenus case by Gaudry and Theriault, including the large prime variant and using a smaller factor base, into the largegenus algorithm of Enge and Gaudry. We extend the analysis in [24] to our new algorithm, allowing us to predict accurately the number of random walk steps required to nd all relations, and to select optimal degree bounds for the factor base. Our second improvement is the adaptation of sieving techniques from Flassenberg and Paulus, and Jacobson to our setting. The new algorithms are applied to concrete problem instances arising from the Weil descent attack methodology for solving the elliptic curve discrete logarithm problem, demonstrating signi cant improvements in practice. 1.
Factoring Small to Medium Size Integers: An Experimental Comparison
, 2010
"... Abstract. We report on our experiments in factoring integers from 50 to 200 bit with the NFS postsieving stage or class group structure computations as potential applications. We implemented, with careful parameter selections, several generalpurpose factoring algorithms suited for these smaller num ..."
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Abstract. We report on our experiments in factoring integers from 50 to 200 bit with the NFS postsieving stage or class group structure computations as potential applications. We implemented, with careful parameter selections, several generalpurpose factoring algorithms suited for these smaller numbers, from Shanks’s square form factorization method to the selfinitializing quadratic sieve, and revisited the continued fraction algorithm in light of recent advances in smoothness detection batch methods. We provide detailed timings for our implementations to better assess their relative range of practical use on current commodity hardware. 1
Sieve with Two Large Primes
"... This paper deals with variations of the Quadratic Sieve integer factoring algorithm. We describe what we believe is the rst implementation of the Hypercube Multiple Polynomial Quadratic Sieve with two large primes, We have used this program to factor many integers with up to 116 digits. Our program ..."
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This paper deals with variations of the Quadratic Sieve integer factoring algorithm. We describe what we believe is the rst implementation of the Hypercube Multiple Polynomial Quadratic Sieve with two large primes, We have used this program to factor many integers with up to 116 digits. Our program appears to be many times faster than the (nonhypercube) Multiple Polynomial Quadratic Sieve with two large primes.
ProjectTeam tanc Algorithmic number theory for cryptology
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Integer Factorization and Computing Discrete Logarithms in Maple
"... have investigated algorithms for integer factorization and computing discrete logarithms. We have implemented a quadratic sieve algorithm for integer factorization in Maple to replace Maple’s implementation of the Morrison ..."
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have investigated algorithms for integer factorization and computing discrete logarithms. We have implemented a quadratic sieve algorithm for integer factorization in Maple to replace Maple’s implementation of the Morrison
INFRASTRUCTURE, ARITHMETIC, AND CLASS NUMBER COMPUTATIONS IN PURELY CUBIC FUNCTION FIELDS OF CHARACTERISTIC AT LEAST 5
, 2009
"... One of the more difficult and central problems in computational algebraic number theory is the computation of certain invariants of a field and its maximal order. In this thesis, we consider this problem where the field in question is a purely cubic function field, K/Fq(x), with char(K) ≥ 5. In add ..."
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One of the more difficult and central problems in computational algebraic number theory is the computation of certain invariants of a field and its maximal order. In this thesis, we consider this problem where the field in question is a purely cubic function field, K/Fq(x), with char(K) ≥ 5. In addition, we will give a divisortheoretic treatment of the infrastructures of K, including a description of its arithmetic, and develop arithmetic on the ideals of the maximal order, O, of K. Historically, the infrastructure, RC, of an ideal class, C ∈ Cl(O) has been defined as a set of reduced ideals in C. However, we extend work of Paulus and Rück [PR99] and Jacobson, Scheidler, and Stein [JSS07b] to define RC as a certain subset of the divisor class group, JK, of a cubic function field, K, specifically, the subset of distinguished divisors whose classes map to C via JK → Cl(O). Our definition of distinguished generalizes the same notion by Bauer for purely cubic function fields of unit rank 0 [Bau04] to those of unit rank 1 and 2 as well. Further, we prove a bijection between RC, as a set of distinguished divisors, and the infrastructure of C defined by “reduced” ideals, as in [Sch00, SS00, Sch01, LSY03, Sch04]. We describe the arithmetic on RC, providing new results on the baby step and giant step operations and generalizing notions of the inverse of a divisor in R [O] from quadratic infrastructures in [JSS07b] to cubic infrastructures. We also give algorithms to