Rotation-Symmetric Boolean Functions – Count and Cryptographic Properties

Abstract

In 1999, Pieprzyk and Qu presented rotation symmetric (RotS) functions as components in the rounds of hashing algorithm. Later, in 2002, Cusick and St ̆anic ̆a presented further advancement in this area. This class of Boolean functions are invariant under circular translation of indices. In this paper, using Burnside’s lemma, we prove that the number of n-variable rotation symmetric Boolean functions is 2gn , where gn = 1/n∑t|nφ(t)2n/t , and φ(.) is the Eulern phi-function. Moreover, we find the number of short and long cycles of elements in Zn/2 having fixed weight, under the RotS action. As a consequence we obtain the number of homogeneous RotS functions having algebraic degree w. Our results make the search space of RotS functions much reduced and we successfully analyzed important cryptographic properties of such functions by executing computer programs. We found that there are exactly 8,48, and 15104, RotS bent functions on 4,6, and 8 variables respectively. Experimental results up to 10 variables show that there is no homogeneous rotation symmetric bent function with degree > 2. Further, we studied the RotS functions on 5,6,7 variables for correlation immunity and propagation characteristics and found some functions with very good cryptographic properties which were not known earlier.