Polynomial Expressions of Carries in p-ary Arithmetics

It is known that any n-variable function on a finite prime field of characteristic p can be expressed as a polynomial over the same field with at most pn monomials. However, it is not obvious to determine the polynomial for a given concrete function. In this paper, we study the concrete polynomial expressions of the carries in addition and multiplication of p-ary integers. For the case of addition, our result gives a new family of symmetric polynomials, which generalizes the known result for the binary case p = 2 where the carries are given by elementary symmetric polynomials. On the other hand, for the case of multiplication of n single-digit integers, we give a simple formula of the polynomial expression for the carry to the next digit using the Bernoulli numbers, and show that it has only (n + 1)(p − 1)/2 + 1 monomials, which is significantly fewer than the worst-case number pn of monomials for general functions. We also discuss applications of our results to cryptographic computation on encrypted data. Remark. The authors are notified that the essential part of our Theorem 2 appears (by a different approach) in: C. Sturtivant, G. S. Frandsen, The Computational Efficacy of Finite-Field Arithmetic, Theoretical Computer Science 112 (1993) 291–309 (see Theorem 9.1(a) and Theorem 11.2 in that paper). The authors deeply thank Akihiro Munemasa for the information. The authors would like to keep this preprint online for reference purposes.