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

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