Rotation symmetric Boolean functions—Count and cryptographic properties

Abstract

Rotation symmetric ( RotS) Boolean functions have been used as components of different cryptosystems. This class of Boolean functions are invariant under circular translation of indices. Using Burnside's lemma it can be seen that the number of n-variable rotation symmetric Boolean functions is 2 g n , where g n = ( 1 / n ) ∑ t | n φ ( t ) 2 n / t , and φ ( . ) is the Euler phi-function. In this paper, we find the number of short and long cycles of elements in F 2 n 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 study RotS bent functions up to 10 variables and observe (experimentally) that there is no homogeneous rotation symmetric bent function having degree > 2 . Further, we studied the RotS functions on 5 , 6 , 7 variables by computer search for correlation immunity and propagation characteristics and found some functions with very good cryptographic properties which were not known earlier.