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

TL;DR

This paper investigates the distribution properties of a subset sum pseudorandom number generator on elliptic curves over finite fields, providing improved results on the uniformity of generated sequences.

Contribution

It generalizes and enhances previous results on the distribution of subset sum pseudorandom sequences on elliptic curves, considering average over all point choices.

Findings

Improved bounds on distribution uniformity.

Generalization of previous distribution results.

Enhanced understanding of pseudorandom sequence behavior.

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 $$ \sum_{j=0}^{r-1} u(n+j)P_j, \qquad n =1,..., N, $$ 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.