Construction of Large Families of Pseudorandom Subsets of the Set {1, 2, ..., N} Using Elliptic Curves

Pseudo-random subsets of the set ${1, 2,ldots, N}$ have many applications in the fields of network security, cryptography and other security issues. Recently, Dartyge and S'{a}rk"{o}zy investigated the measures--the well distribution measure and the correlation measure of order $k$--of pseudorandomness of subsets of the set ${1, 2,ldots, N}$, and they presented several constructive examples for subsets with strong pseudorandom properties when $N$ is a prime number. In this article, we present a construction of pseudorandom subsets by using elliptic curves over finite fields and estimate their pseudorandom measures. Exponential sums play an important role in the proofs.