COMPUTING THE NUMBER OF POINTS ON GENUS 3 HYPERELLIPTIC CURVES OF TYPE Y2= X7+ aX OVER FINITE PRIME FIELDS

Abstract

In this paper, we present an algorithm for computing the number of points on the Jacobian varieties of genus 3 hyperelliptic curves of type <TEX>$y^2=x^7+ax$</TEX> over finite prime fields. The problem of determining the group order of the Jacobian varieties of algebraic curves defined over finite fields is important not only arithmetic geometry but also curve-based cryptosystems in order to find a secure curve. Based on this, we provide the explicit formula of the characteristic polynomial of the Frobenius endomorphism of the Jacobian variety of hyperelliptic curve <TEX>$y^2=x^7+ax$</TEX> over a finite field <TEX>$\mathbb{F}_p$</TEX> with <TEX>$p{\equiv}1$</TEX> modulo 12. Moreover, we also introduce some implementation results by using our algorithm.