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

  • İsa SertkayaEmail author
  • Ali Doğanaksoy
  • Osmanbey UzunkolEmail author
  • Mehmet Sabır Kiraz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9061)


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 



We would like to thank the anonymous reviewers for their valuable comments on the preliminary version of this work. Third author is partly supported by a joint research project funded by Bundesministerium für Bildung und Forschung (BMBF), Germany (01DL12038) and TÜBİTAK Turkey (TBAG-112T011).


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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

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