Algorithms for solving linear systems over cyclotomic fields

We consider the problem of solving a linear system Ax = b over a cyclotomic field. Cyclotomic fields are special in that we can easily find a prime p for which the minimal polynomial m(z) for the field factors into a product of distinct linear factors. This makes it possible to develop fast modular algorithms. We give two output sensitive modular algorithms, one using multiple primes and Chinese remaindering, the other using linear p-adic lifting. Both use rational reconstruction to recover the rational coefficients in the solution vector. We also give a third algorithm which computes the solutions as ratios of two determinants modulo m(z) using Chinese remaindering only. Because this representation is d = deg m(z) times more compact in general, we can compute it the fastest. We have implemented the algorithms in Maple. Our benchmarks show that the third method is fastest on random inputs, but on real inputs arising from problems in computational group theory, the first two methods are faster because the solutions have small rational coefficients. (C) 2010 Elsevier Ltd. All rights reserved.