On affine (non)equivalence of Boolean functions

In this paper we construct a multiset S(f) of a Boolean function f consisting of the weights of the second derivatives of the function f with respect to all distinct two-dimensional subspaces of the domain. We refer to S(f) as the second derivative spectrum of f. The frequency distribution of the weights of these second derivatives is referred to as the weight distribution of the second derivative spectrum. It is demonstrated in this paper that this weight distribution can be used to distinguish affine nonequivalent Boolean functions. Given a Boolean function f on n variables we present an efficient algorithm having O(n22n) time complexity to compute S(f). Using this weight distribution we show that all the 6-variable affine nonequivalent bents can be distinguished. We study the subclass of partial-spreads type bent functions known as PSap type bents. Six different weight distributions are obtained from the set of PSap bents on 8-variables. Using the second derivative spectrum we show that there exist 6 and 8 variable bent functions which are not affine equivalent to rotation symmetric bent functions. Lastly we prove that no non-quadratic Kasami bent function is affine equivalent to Maiorana–MacFarland type bent functions.

[1]  Claude Carlet.  Partially-bent functions , 1993, Des. Codes Cryptogr..

[2]  Timo Neumann,et al.  BENT FUNCTIONS , 2006 .

[3]  Huanguo Zhang,et al.  Analysis of affinely equivalent Boolean functions , 2007, Science in China Series F: Information Sciences.

[4]  Bart Preneel,et al.  Classification of Boolean Functions of 6 Variables or Less with Respect to Some Cryptographic Properties , 2005, ICALP.

[5]  Hans Dobbertin,et al.  A Survey of Some Recent Results on Bent Functions , 2004, SETA.

[6]  Claude Carlet,et al.  Two New Classes of Bent Functions , 1994, EUROCRYPT.

[7]  Anne Canteaut,et al.  Finding nonnormal bent functions , 2006, Discret. Appl. Math..

[8]  O. S. Rothaus,et al.  On "Bent" Functions , 1976, J. Comb. Theory, Ser. A.

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

[10]  Claude Carlet,et al.  Partially-bent functions , 1992, Des. Codes Cryptogr..

[11]  A. Canteaut,et al.  Decomposing bent functions , 2002, Proceedings IEEE International Symposium on Information Theory,.

[12]  Harald Niederreiter,et al.  Introduction to finite fields and their applications: Preface , 1994 .

[13]  Bart Preneel,et al.  Cryptographic hash functions , 2010, Eur. Trans. Telecommun..

[14]  Xiang-dong Hou.  Cubic bent functions , 1998, Discret. Math..

[15]  Sugata Gangopadhyay,et al.  On Kasami Bent Functions , 2008, IACR Cryptol. ePrint Arch..

[16]  Pantelimon Stanica,et al.  Rotation Symmetric Boolean Functions -; Count and Cryptographic Properties , 2003, Electron. Notes Discret. Math..

[17]  Hans Dobbertin,et al.  Construction of Bent Functions and Balanced Boolean Functions with High Nonlinearity , 1994, FSE.

[18]  J. Dillon.  Elementary Hadamard Difference Sets , 1974 .

[19]  Frederik Armknecht,et al.  Efficient Computation of Algebraic Immunity for Algebraic and Fast Algebraic Attacks , 2006, EUROCRYPT.

[20]  Xiang-dong Hou.  GL(m, 2) acting on R(r, m)/R(r-1, m) , 1996, Discret. Math..

[21]  Hans Dobbertin,et al.  New cyclic difference sets with Singer parameters , 2004, Finite Fields Their Appl..