Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials

Abstract

Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to the genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered the simplest model. Explicit formulas defining reduced divisors for some particular cases are found. The reduced divisors are obtained in the form of a solution of the Jacobi inversion problem, which provides a 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 compared with the known in cryptography, and is extended to divisors of arbitrary degrees compared with the known in the theory of hyperelliptic functions.