Linear and sublinear time algorithms for the basis of abelian groups

It is well known that every finite abelian group G can be represented as a direct product of cyclic groups: G ? G 1×G 2× ? ×G t , where each G i is a cyclic group of order p j for some prime p and integer j ? 1. If a i generates the cyclic group of G i , i = 1,2, ? , t, then the elements a 1,a 2, ? , a t are called a basis of G. We show a randomized algorithm such that given a set of generators M = {x 1, ? , x k } for an abelian group G and the prime factorization of order ord(x i ) (i = 1, ? , k), it computes a basis of G in $O(|M|(\log n)^2+\sum_{i=1}^t n_ip_i^{n_i/2})$ time, where n = |G| has prime factorization $p_1^{n_1}p_2^{n_2}\cdots p_t^{n_t}$ (which is not a part of input). This generalizes Buchmann and Schmidt's algorithm that takes $O(|M|\sqrt{|G|})$ time. In another model, all elements in an abelian group are put into a list as a part of input. We obtain an O(n) time deterministic algorithm and a subliner time randomized algorithm for computing a basis of an abelian group.