Affine Equivalency and Nonlinearity Preserving Bijective Mappings over \(\mathbb {F}_2\)

Conference paper

DOI: 10.1007/978-3-319-16277-5_7

Volume 9061 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Sertkaya İ., Doğanaksoy A., Uzunkol O., Kiraz M.S. (2015) Affine Equivalency and Nonlinearity Preserving Bijective Mappings over \(\mathbb {F}_2\). In: Koç Ç., Mesnager S., Savaş E. (eds) Arithmetic of Finite Fields. WAIFI 2014. Lecture Notes in Computer Science, vol 9061. Springer, Cham


We first give a proof of an isomorphism between the group of affine equivalent maps and the automorphism group of Sylvester Hada-mard matrices. Secondly, we prove the existence of new nonlinearity preserving bijective mappings without explicit construction. Continuing the study of the group of nonlinearity preserving bijective mappings acting on \(n\)-variable Boolean functions, we further give the exact number of those mappings for \(n\,\le \,6\). Moreover, we observe that it is more beneficial to study the automorphism group of bijective mappings as a subgroup of the symmetric group of the \(2^{n}\) dimensional \(\mathbb {F}_{2}\)-vector space due to the existence of non-affine mapping classes.


Cryptographic Boolean functions Affine equivalence Nonlinearity preserving mappings Sylvester Hadamard matrices 

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Mathematical and Computational SciencesTÜBİTAK BİLGEM UEKAEGebzeTurkey
  3. 3.Department of MathematicsMiddle East Technical UniversityAnkaraTurkey