On the Complexity of Computing Units in a Number Field

Given an algebraic number field K, such that [K : Q] is constant, we show that the problem of computing the units group O∗ K is in the complexity class SPP. As a consequence, we show that principal ideal testing for an ideal in OK is in SPP. Furthermore, assuming the GRH, the class number of K, and a presentation for the class group of K can also be computed in SPP. A corollary of our result is that solving PELL S EQUATION, recently shown by Hallgren [12] to have a quantum polynomial-time algorithm, is also in SPP.

• Type: Conference
• Year: 2004
• Where: ANTS
• Authors: Vikraman Arvind, Piyush P. Kurur
• Keywords: Algorithms, ANTS 2004, Complexity Class Spp, Number field K, Principal Ideal Testing