Construction of large families of pseudo-random binary sequences

In a series of papers Mauduit and S arkozy (partly with coauthors) studied nite pseudorandom binary sequences. They showed that the Legendre symbol forms a \good" pseudorandom sequence, and they also tested other sequences for pseudorandomness, however, no large family of \good" pseudorandom sequences has been found yet. In this paper, a large family of this type is constructed. Again, the construction is related to the Legendre symbol. Moreover, by using elliptic curves large families of binary sequences are constructed. It is expected that these sequences form \good" pseudorandom binary sequences. 1991 Mathematics Subject Classi cation : 11 K 45.