On the elliptic curve endomorphism generator

TL;DR

This paper investigates the properties of sequences generated by elliptic curve endomorphisms, focusing on their distribution and linear complexity, which are crucial for cryptographic applications.

Contribution

It introduces a new framework for analyzing sequences generated by elliptic curve endomorphisms and studies their distribution and linear complexity properties.

Findings

Sequences exhibit certain distribution patterns.

Linear complexity of sequences is characterized.

Implications for cryptographic security are discussed.

Abstract

For an elliptic curve $E$ over a finite field we define the point sequence $(P_n)$ recursively by $P_n=\vartheta (P_{n-1})=\vartheta ^n(P_0)$ with an endomorphism $\vartheta \in\mathrm{End}(E)$ and with some initial point $P_0$ on $E$. We study the distribution and the linear complexity of sequences obtained from $(P_n)$.