Explicit Formulas for Real Hyperelliptic Curves of Genus 2 in Affine Representation

Abstract

In this paper, we present for the first time efficient explicit formulas for arithmetic in the degree 0 divisor class group of a real hyperelliptic curve. Hereby, we consider real hyperelliptic curves of genus 2 given in affine coordinates for which the underlying finite field has characteristic > 3. These formulas are much faster than the optimized generic algorithms for real hyperelliptic curves and the cryptographic protocols in the real setting perform almost as well as those in the imaginary case. We provide the idea for the improvements and the correctness together with a comprehensive analysis of the number of field operations. Finally, we perform a direct comparison of cryptographic protocols using explicit formulas for real hyperelliptic curves with the corresponding protocols presented in the imaginary model.Keywordshyperelliptic curvereduced divisorinfrastructure and distanceCantor’s algorithmexplicit formulasefficient implementationcryptographic key exchange