Fast component-by-component construction for (non-)primes ?

The component-by-component construction algorithm for rank-1 lattices can be formulated elegantly as a repeated matrix-vector product. As was shown in an earlier paper, this matrix-vector product can be done in time O(n log(n)) for n prime. Here we extend this result to general n using facts from algebra to obtain a construction cost of O(sn log(n)) for a rank-1 lattice in s dimensions with n points. As was the case for n prime, the main calculation cost is significantly reduced by using fast Fourier transforms in the matrix-vector calculation. The number of fast Fourier transforms is dependent on the number of divisors of n and the number of prime factors of n. It is believed that the intrinsic structure present in rank-1 lattices and exploited by this fast construction method will deliver new insights in the applicability of these lattices.