Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials

TL;DR

This paper presents explicit formulas and an efficient polynomial-based algorithm for reducing and adding divisors on hyperelliptic curves, facilitating computations of Abelian functions for cryptography and algebraic geometry.

Contribution

It introduces a novel polynomial arithmetic algorithm for divisor reduction and addition on hyperelliptic curves, extending existing methods to arbitrary divisor degrees.

Findings

Effective reduction algorithm using only polynomial operations

Explicit formulas for reduced divisors in specific cases

Extended addition algorithm for divisors of arbitrary degrees

Abstract

Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered as the simplest model. Explicit formulas defining reduced divisors for some particular cases are found. The reduced divisors are obtained in the form of solution of the Jacobi inversion problem which provides the way of computing Abelian functions on arbitrary non-special divisors. An effective reduction algorithm is proposed, which has the advantage that it involves only arithmetic operations on polynomials. The proposed addition algorithm contains more details comparing with the known in cryptography, and is extended to divisors of arbitrary degrees comparing with the known in the theory of hyperelliptic functions.