On the Equivalence of Nonlinear Functions

Recently, many new almost perfect nonlinear (APN) and almost bent (AB) functions have been constructed. These functions F n 2 → F n 2 play an important role in cryptography. In this article, we will summarize different concepts of equivalence between these functions, and discuss some invariants. Two codes can be associated with APN and AB functions. This is useful to distinguish functions up to equivalence. We give a short proof about the dimension of one of these codes. We slightly extend the known concepts of equivalence to the more general case of functions F n 2 → F m 2 . Moreover, we show that CCZ equivalence is the same as extended affine equivalence if F is a vectorial bent function.

[1]  Bernhard Schmidt,et al.  Characters and Cyclotomic Fields in Finite Geometry , 2002 .

[2]  Claude Carlet,et al.  Constructing new APN functions from known ones , 2009, Finite Fields Their Appl..

[3]  C. Colbourn,et al.  The CRC handbook of combinatorial designs , edited by Charles J. Colbourn and Jeffrey H. Dinitz. Pp. 784. $89.95. 1996. ISBN 0-8493-8948-8 (CRC). , 1997, The Mathematical Gazette.

[4]  Kaisa Nyberg,et al.  Perfect Nonlinear S-Boxes , 1991, EUROCRYPT.

[5]  Eimear Byrne,et al.  Quadratic Almost Perfect Nonlinear Functions With Many Terms , 2007, IACR Cryptol. ePrint Arch..

[6]  Claude Carlet,et al.  New classes of almost bent and almost perfect nonlinear polynomials , 2006, IEEE Transactions on Information Theory.

[7]  Mitsuru Matsui,et al.  Linear Cryptanalysis Method for DES Cipher , 1994, EUROCRYPT.

[8]  Cunsheng Ding,et al.  Highly nonlinear mappings , 2004, J. Complex..

[9]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[10]  Eimear Byrne,et al.  New families of quadratic almost perfect nonlinear trinomials and multinomials , 2008, Finite Fields Their Appl..

[11]  Claude Carlet,et al.  Codes, Bent Functions and Permutations Suitable For DES-like Cryptosystems , 1998, Des. Codes Cryptogr..

[12]  Claude Carlet,et al.  On CCZ-equivalence and its use in secondary constructions of bent functions , 2009, IACR Cryptol. ePrint Arch..

[13]  Hanfried Lenz,et al.  Design theory , 1985 .

[14]  K. T. Arasu,et al.  Groups, Difference Sets, and the Monster: Proceedings of a Special Research Quarter at The Ohio State University, Spring 1993 , 1996 .

[15]  Claude Carlet,et al.  Boolean Functions for Cryptography and Error-Correcting Codes , 2010, Boolean Models and Methods.

[16]  Claude Carlet,et al.  Two Classes of Quadratic APN Binomials Inequivalent to Power Functions , 2008, IEEE Transactions on Information Theory.

[17]  Alexander Pott,et al.  A new APN function which is not equivalent to a power mapping , 2005, IEEE Transactions on Information Theory.

[18]  A. T. Butson,et al.  Relative difference sets , 1966 .

[19]  Alexander Pott,et al.  A new almost perfect nonlinear function which is not quadratic , 2008, Adv. Math. Commun..

[20]  Bernhard Schmidt,et al.  On (pa, pb, pa, pa-b)-Relative Difference Sets , 1997 .

[21]  Claude Carlet,et al.  Classes of Quadratic APN Trinomials and Hexanomials and Related Structures , 2008, IEEE Transactions on Information Theory.

[22]  Eli Biham,et al.  Differential cryptanalysis of DES-like cryptosystems , 1990, Journal of Cryptology.

[23]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..

[24]  Serge Vaudenay,et al.  Links Between Differential and Linear Cryptanalysis , 1994, EUROCRYPT.

[25]  Gregor Leander,et al.  On the classification of APN functions up to dimension five , 2008, Des. Codes Cryptogr..

[26]  Alexander Pott,et al.  Nonlinear functions in abelian groups and relative difference sets , 2004, Discret. Appl. Math..

[27]  A. Pott.  A survey on relative difference sets , 1996 .