Enhanced Count of Balanced Symmetric Functions and Balanced Alternating Functions

摘要

This paper extends and improves known results on finite symmetric functions: f : E_a~n → E_b, to any a and any b, where E_a (respectively Eb) stands for the set of a (respectively b) elements. It also generalises them to alternating functions. The main motivation is that for actual implementation in ciphers a and b are required to be powers of 2 instead of a prime. Another driver is the ease of implementation of these functions especially for very large n. Search algorithms for these functions were defined and implemented with relevant improvements and significant new counting results. Our main theorem gives a new enumeration formula thanks to two theoretical enhancements. We provide exact enumeration values where only lower bounds were previously known. The non existence of balanced finite symmetric or alternating functions is proven under some conditions on o, b, and n. We also exhibit new results for new values of a and b, giving very large numbers of functions.