Abstract
Hafner and McCurley described a subexponential time algorithm to compute the ideal class group of a quadratic field, which was generalized to families of fixed degree number fields by Buchman. The main ingredient of this method is a subexponential time algorithm to derive relations between primes of norm bounded by a subexponential value. Besides ideal class group computation, this was successfully used to evaluate isogenies, compute endomorphism rings, solve the discrete logarithm problem in the class group and find a generator of a principal ideal. In this paper, we present a generalization of the relation search to classes of number fields with degree growing to infinity.
Highlights
Let K = Q(θ) be a number field of degree n and maximal order OK, a be an ideal of an order O ⊆ OK, and a bound B > 0
Buchmann [6] generalized a result of Hafner and McCurley [16] to prove that the ideal class group and the unit group of the maximal order of classes of number fields of fixed degree and discriminant ∆ growing to infinity could be computed in time L∆(1/2, c) where c > 0 is a constant and L∆(a, b) := eb log |∆|a log log |∆|1−a
We present an algorithm relying on BKZ-reductions with subexponential complexity on any infinite class of orders and a q-descent algorithm working on restricted classes
Summary
Buchmann [6] generalized a result of Hafner and McCurley [16] to prove that the ideal class group and the unit group of the maximal order of classes of number fields of fixed degree and discriminant ∆ growing to infinity could be computed in time L∆(1/2, c) where c > 0 is a constant and L∆(a, b) := eb log |∆|a log log |∆|1−a. His proof relies on the capacity to derive relations of the form (1) for B = L∆(1/2, c1) in time L∆(1/2, c2) for constants c1, c2 > 0. It is the probability of the smoothness of an ideal which rules the run time of an algorithm, and in the other case, it is the probability of smoothness of an element, which is much less understood, in particular due to the units of O
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