Algorithms for rotation symmetric Boolean functions

Rotation Symmetric Boolean Functions (RSBFs) are of immense importance as building blocks of cryptosystems. This class of Boolean functions are invariant under circular translation of indices. It is known that, for n-variable RSBF functions, the associated set of all possible input n-bit strings can be divided into a number of subsets (called partitions or orbits), where every element (n-bit string) of a subset can be obtained by simply rotating the string of bits of some other element of the same subset. In this paper, for a given value of n, we propose algorithms for the generation of these partitions of all possible n-bit strings, each partition corresponding to a specific n-variable RSBF and its associated circular translations. These partitions can then be used to generate all possible n-variable RSBFs. The proposed algorithms are implemented for a maximum value of n = 41.

[1]  Josef Pieprzyk,et al.  Rotation-Symmetric Functions and Fast Hashing , 1998, J. Univers. Comput. Sci..

[2]  Niraj K. Jha,et al.  Switching and Finite Automata Theory: Frontmatter , 2009 .

[3]  Subhamoy Maitra,et al.  Construction of Rotation Symmetric Boolean Functions on Odd Number of Variables with Maximum Algebraic Immunity , 2007, AAECC.

[4]  John A. Clark,et al.  Results on Rotation Symmetric Bent and Correlation Immune Boolean Functions , 2004, FSE.

[5]  Tsutomu Sasao,et al.  Logic functions for cryptography - A tutorial , 2009 .

[6]  Selçuk Kavut,et al.  Enumeration of 9-Variable Rotation Symmetric Boolean Functions Having Nonlinearity > 240 , 2006, INDOCRYPT.

[7]  Narsingh Deo,et al.  Graph Theory with Applications to Engineering and Computer Science , 1975, Networks.

[8]  Chao Li,et al.  On the number of rotation symmetric Boolean functions , 2010, Science China Information Sciences.

[9]  Palash Sarkar,et al.  Construction of Nonlinear Boolean Functions with Important Cryptographic Properties , 2000, EUROCRYPT.

[10]  Pantelimon Stanica,et al.  Rotation symmetric Boolean functions - Count and cryptographic properties , 2003, Discret. Appl. Math..

[11]  Massoud Pedram,et al.  Power minimization in IC design: principles and applications , 1996, TODE.

[12]  Yifa Li,et al.  Proof of a conjecture about rotation symmetric functions , 2011, Discret. Math..

[13]  Anne Canteaut,et al.  Symmetric Boolean functions , 2005, IEEE Transactions on Information Theory.