On the Distribution of the Subset Sum Pseudorandom Number Generator on Elliptic Curves

Abstract

Given a prime $p$, an elliptic curve $\E/\F_p$ over the finite field $\F_p$ of $p$ elements and a binary \lrs\ $\(u(n)\)_{n =1}^\infty$ of order~$r$, we study the distribution of the sequence of points $$\sum _{j=0}^{r-1}u(n+j)P_j,n=1,...,N,$$ on average over all possible choices of $\F_p$-rational points $P_1,...,P_r$ on~$\E$. For a sufficiently large $N$ we improve and generalise a previous result in this direction due to E.~El~Mahassni.