Efficient Computation of the Number of Solutions of the Linear Diophantine Equation of Frobenius with Small Coefficients

In this paper we present a novel approach for computing the number of solutions of the linear diophantine equation of Frobenius a1·x1 + ... + aN·xN = T when the coefficients a1, ..., aN are small. The proposed algorithm has a time complexity of the order of O(N·S·log(T)), where S=a1+...+aN. The algorithm can also be implemented to run in O(S·log(S)·log(T)+N·S) time, which is more efficient when log(S)<N. Note that an elementary operation in these cases consists of an arithmetic operation (e.g. addition, multiplication) applied on numbers of the order of magnitude of the result, which may be exponential in both N and S. A common situation consists of computing the result modulo a given number P, in which case the arithmetic operation is applied on numbers of the same order of magnitude as P.