On the linear complexity profile of some sequences derived from elliptic curves

TL;DR

This paper investigates the linear complexity profiles of sequences derived from elliptic curves over finite fields, introducing new methods using Edwards coordinates and analyzing generalized elliptic curve power generators.

Contribution

It extends previous work by Hess and Shparlinski, providing new results on linear complexity profiles using Edwards coordinates and exploring sequences from elliptic curve power generators.

Findings

Identified large families of functions with high linear complexity profiles

Extended analysis to sequences from elliptic curve power generators

Utilized Edwards coordinates to broaden applicability of results

Abstract

For a given elliptic curve $\mathbf{E}$ over a finite field of odd characteristic and a rational function $f$ on $\mathbf{E}$ we first study the linear complexity profiles of the sequences $f(nG)$, $n=1,2,\dots$ which complements earlier results of Hess and Shparlinski. We use Edwards coordinates to be able to deal with many $f$ where Hess and Shparlinski's result does not apply. Moreover, we study the linear complexities of the (generalized) elliptic curve power generators $f(e^nG)$, $n=1,2,\dots$. We present large families of functions $f$ such that the linear complexity profiles of these sequences are large.