Transitive q-Ary Functions over Finite Fields or Finite Sets: Counts, Properties and Applications

Abstract

To implement efficiently and securely good non-linear functions with a very large number of input variables is a challenge. Partially symmetric functions such as transitive functions are investigated to solve this issue. Known results on Boolean symmetric functions are extended both to transitive functions and to q-ary functions (on any set of q elements including finite fields GF(q) for any q). In a special case when the number of variables is n = p k with p prime, an extension of Lucas’ theorem provides new counting results and gives useful properties on the set of transitive functions. Results on balanced transitive q-ary functions are given. Implementation solutions are suggested based on q-ary multiple-valued decision diagrams and examples show simple implementations for these kind of symmetric functions. Applications include ciphers design and hash functions design but also search for improved covering radius of codes.KeywordsSymmetric functions(sharply) t-transitive functionsbalanced functionsfunctions over finite fieldshardware and software implementation