In this paper, we introduce a new approach to the generation of binary sequences by applying trace functions to elliptic curves over GF(2^m). We call these sequences elliptic curve pseudorandom sequences (EC-sequence). We determine their periods, distribution of zeros and ones, and linear spans for a class of EC-sequences generated from supersingular curves. We exhibit a class of EC-sequences which has half period as a lower bound for their linear spans. EC-sequences can be constructed algebraically and can be generated efficiently in software or hardware by the same methods that are used for implementation of elliptic curve public-key cryptosystems.