Linear symmetries of Boolean functions

Abstract

In this note we study the linear symmetry group LS ( f ) of a Boolean function f of n variables, that is, the set of all σ ∈ GL n ( 2 ) which leave f invariant, where GL n ( 2 ) is the general linear group on the field of two elements. The main problem is that of concrete representation: which subgroups G of GL n ( 2 ) can be represented as G = LS ( f ) for some n-ary k-valued Boolean function f. We call such subgroups linearly representable. The main results of the note may be summarized as follows: We give a necessary and sufficient condition that a subgroup of GL n ( 2 ) is linearly representable and obtain some results on linear representability of its subgroups. Our results generalize some theorems from P. Clote and E. Kranakis [SIAM J. Comput. 20 (1991) 553–590]; A. Kisielewicz [J. Algebra 199 (1998) 379–403].