Linear complexity of some sequences derived from hyperelliptic curves of genus 2

TL;DR

This paper investigates sequences derived from genus 2 hyperelliptic curves over finite fields, demonstrating they have high linear complexity, which is important for cryptographic applications.

Contribution

It introduces a hyperelliptic analogue of the congruential generator and proves that genus 2 curves generate sequences with large linear complexity.

Findings

Sequences from genus 2 hyperelliptic curves have large linear complexity.

The hyperelliptic analogue of the congruential generator is effective.

High linear complexity suggests cryptographic strength.

Abstract

For a given hyperelliptic curve $C$ over a finite field with Jacobian $J_C$, we consider the hyperelliptic analogue of the congruential generator defined by $W_n=W_{n-1}+D$ for $n\geq 1$ and $D,W_0\in J_C$. We show that curves of genus 2 produce sequences with large linear complexity.