Linear symmetries of Boolean functions

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 ∈ GLn(2) which leave f invariant, where GLn(2) is the general linear group on the field of two elements. The main problem is that of concrete representation: which subgroups G of GLn(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 ofGLn(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]. © 2005 Elsevier B.V. All rights reserved.