Abstract
We investigate improvements to the algorithm for the computationof ideal class groups described by Jacobson in the imaginary quadratic case.These improvements rely on the large prime strategy and a new method forperforming the linear algebra phase. We achieve a significant speed-up and areable to compute ideal class groups with discriminants of 110 decimal digits inless than a week.
Highlights
Given a fundamental discriminant ∆, it is known that the corre√sponding ideal class group Cl(∆) of the order O∆ of discriminant ∆ in K = Q( ∆) is a finite abelian group that can be decomposed as Cl(∆)Z/diZ, i where the divisibility condition di|di+1 holds
Some key-exchange protocols relying on the difficulty of solving the discrete logarithm problem (DLP) in imaginary quadratic orders have been proposed [3, 9] and solving instances of the DLP is closely related to finding the group structure of Cl(∆)
In 1968 Shanks [18] proposed an algorithm relying on the baby-step giant-step method in order to compute the structure of the ideal class group of an imaginary quadratic number field in time O |∆|1/4+, or O |∆|1/5+ under the extended
Summary
In 1968 Shanks [18] proposed an algorithm relying on the baby-step giant-step method in order to compute the structure of the ideal class group of an imaginary quadratic number field in time O |∆|1/4+ , or O |∆|1/5+ under the extended. Buchman and Dullmann [2] computed class groups with discriminants of around 50 decimal digits using an implementation of this algorithm An improvement of this method was published by Jacobson in 1999 [10]. He achieved a significant speed-up by using sieving strategies to generate the matrix of relations. He was able to compute the structure of class groups of discriminants having up to 90. Our approach is based on that of Jacobson, using new techniques to accelerate both the sieving phase and the linear algebra phase; we have obtained the group structure of class groups of 110 decimal digit discriminants
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