Explicit formulas for real hyperelliptic curves of genus 2 in affine representation

Abstract

We present a complete set of efficient explicit formulas for arithmeticin the degree $0$ divisor class group of a genus two real hyperellipticcurve given in affine coordinates. In addition to formulas suitable forcurves defined over an arbitrary finite field, we give simplified versionsfor both the odd and the even characteristic cases. Formulas for babysteps, inverse baby steps, divisor addition, doubling, and specialcases such as adding a degenerate divisor are provided, withvariations for divisors given in reduced and adapted basis. Wedescribe the improvements and the correctness together with acomprehensive analysis of the number of field operations for eachoperation. Finally, we perform a direct comparison of cryptographicprotocols using explicit formulas for real hyperelliptic curves withthe corresponding protocols presented in the imaginary model.