  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.

Algebraic Geometry;Combinatorics;Number Theory