# Linear symmetries of Boolean functions

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 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). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Applied Mathematics Elsevier

# Linear symmetries of Boolean functions

Discrete Applied Mathematics, Volume 149 (1) – Aug 1, 2005
8 pages

/lp/elsevier/linear-symmetries-of-boolean-functions-OQEFqW54lm
Publisher
Elsevier
ISSN
0166-218X
D.O.I.
10.1016/j.dam.2005.02.008
Publisher site
See Article on Publisher Site

### 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).

### Journal

Discrete Applied MathematicsElsevier

Published: Aug 1, 2005

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