Polynomial Expressions of Carries in p-ary Arithmetics
Abstract
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 $p^n$ 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 $p^n$ of monomials for general functions. We also discuss applications of our results to cryptographic computation on encrypted data.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2015
- DOI:
- 10.48550/arXiv.1506.02742
- arXiv:
- arXiv:1506.02742
- Bibcode:
- 2015arXiv150602742K
- Keywords:
-
- Mathematics - Combinatorics;
- Computer Science - Cryptography and Security;
- Computer Science - Information Theory;
- Mathematics - Number Theory;
- 11T06 (primary);
- 05E05;
- 68R05;
- 94A60
- E-Print:
- (v2) Improved results and new observations (v3) The authors are notified that our main theorem (Theorem 2) appears (by a different approach) in [C. Sturtivant, G. S. Frandsen: Theoretical Computer Science 112 (1993) 291-309]. The authors would like to keep this preprint online for reference purposes