Some remarks concerning the complexity of computing class groups of quadratic fields

AbstractLet O be an order of any quadratic number field. In this paper we show that under the assumption of the generalized Riemann hypothesis the following decision problems are in NP ⊃ co-NP: 1.1. Is a given ideal A of O principal?2.2. Given ideals A1, … , Aκ of O, do their equivalence classes generate the class group of O?3.3. Given ideals A1, … , Aκ of O, is the class group of O the direct product of the cyclic subgroups generated by the equivalence classes of the A1

Some Remarks Concerning the Complexity of Computing Class Groups of Quadratic Fields

Let O be an order of a quadratic number field. In this paper we show that under the assumption of the generalized Riemann hypothesis the following decision problems are in NP " co-NP: 1. Is a given ideal A of O principal? 2. Given ideals A 1 ; : : : ; A k of O, do their equivalence classes generate the class group of O. 3. Given ideals A 1 ; : : : ; A k of O, do their equivalence classes form a basis for the class group of O? 1 Introduction Let \Delta be a rational integer which is not a perfect square, \Delta j 0; 1 mod 4. Then O = Z + Z \Delta + p \Delta 2 is the quadratic order of discriminant \Delta. In McCurley [3] and Buchmann/Williams [1] it was shown that the following decision problems belong to the complexity class NP. (P) Is a given ideal A in O principal? (h) Is h 0 2 Z1 equal to the class number h of O? We remark that (h) could be proved to be in NP only under the assumption of the the generalized Rieman Hypothesis (GRH). In addition, we consider in this paper t..