Affine equivalence for cubic rotation symmetric Boolean functions with [formula omitted] variables

A recursive formula for weights of Boolean rotation symmetric functions

Two Boolean functions are affine equivalent if one can be obtained from the other by applying an affine transformation to the input variables. For a long time, there have been efforts to investigate the affine equivalence of Boolean functions. Due to the complexity of the general problem, only affine equivalence under certain groups of permutations is usually considered. Boolean functions which are invariant under the action of cyclic rotation of the input variables are known as rotation symmetric (RS) Boolean functions. Due to their speed of computation and the prospect of being good cryptographic Boolean functions, this class of Boolean functions has received a lot of attention from cryptographic researchers. In this paper, we study affine equivalence for the simplest rotation symmetric Boolean functions, called MRS functions, which are generated by the cyclic permutations of a single monomial. Using Pólya’s enumeration theorem, we compute the number of equivalence classes, under certain large groups of permutations, for these MRS functions in any number

On the weight and nonlinearity of homogeneous rotation symmetric Boolean functions of degree 2

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

Rotation symmetric Boolean functions (RSBFs) which are invariant under circular translation of indices have been used as components of different cryptosystems. In this paper, we study the construction of RSBFs with maximum algebraic immunity. First, a new construction of RSBFs on odd number of variables with maximum possible Algebraic Immunity is given. Then by using the relationship between some flats and support of a n -variables Boolean function f , we prove that a construction of RSBFs on even number of variables has maximum possible Algebraic Immunity. Furthermore, we study the nonlinearity of functions by our construction.

Rotation symmetric Boolean functions (RSBFs) have been used as components of different cryptosystems. This class of functions are invariant under circular translation of indices. In this paper, we investigated balanced RSBFs and 1st order correlation immune RSBFs. Based on constructive techniques, we give an accurate enumeration formula for n-variable balanced RSBFs when n is a power of a prime. Furthermore, an original and efficient method to enumerate all n-variable (n prime) 1st order correlation-immune functions is presented. The exact number of 1st order correlation immune RSBFs with 11 variables is 6925047156550478825225250374129764511077684773805520800 and the number of 13 variables has 189 digits. Then for more variables, we also provide a significant lower bound on the number of 1st order correlation immune RSBFs.

Secondary constructions of RSBFs with good cryptographic properties

Rotation symmetric Boolean functions (RSBFs) are used widely in symmetric cryptography. In this paper, a systematic construction of balanced odd-variable RSBFs satisfying strict avalanche criterion is proposed. Hence, a class of $(6k+3)$ -variable resilient RSBFs satisfying strict avalanche criterion is also presented for any $k\geq 2$ . Some of the obtained RSBFs have many other good cryptographic properties at the same time, that is, optimal algebraic degree, good global avalanche characteristics, high nonlinearity and nonexistence of nonzero linear structures. Moreover, we obtain some count results of RSBFs satisfying strict avalanche criterion. This is the first time that the autocorrelation properties of RSBFs are investigated systemically.

Rotation symmetric functions have been used as components of different cryptosystems. Functions in this class are invariant under circular translation of indices. In this paper, we will do some enumeration on homogeneous rotation symmetric functions over F p . A formula for counting homogeneous rotation symmetric functions over F p is presented when gcd(n ,d ) is a prime power, where n is the number of input variables and d is the algebraic degree of the function, which demonstrates that we solved one of the problems in [7].

On the matrix of rotation symmetric Boolean functions

In this paper we propose a construction and implementation for a class of rotation-symmetric S-boxes (RSSBs) with good cryptographic properties and improved DPA resilience. The S-boxes are constructed from rotation symmetric Boolean functions (RSBFs) and these RSBFs were searched to avoid any fixed points in the RSSB maps. In literature, search of RSSBs included those consisting of fixed points which is considered to be a weakness in the S-box constructions. We present some new properties of RSSBs and find that the search space of RSSBs with fixed points is of exponential order. We also present the hardware architecture of the RSSBs with no fixed points and the corresponding implementations on Xilinx Virtex-5 FPGA device on SASEBO-GII development board and perform a correlation analysis DPA of AES which include these RSSBs. The RSSBs from the proposed class when incorporated in AES, required more power traces compared to the AES containing Rijndael S-box which indicates that the DPA resilience of the proposed RSSBs is higher than that of the AES Rijndael S-box. Also we present the correlation analysis DPA results on the look-up table, distributed memory and block memory based implementations of some of the RSSBs from the proposed class and compare the results with those of the respective implementations of AES-128 Rijndael S-box.

Balanced [formula omitted]-variable rotation symmetric Boolean functions with optimal algebraic immunity, good nonlinearity, and good algebraic degree

Recently, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rotation Symmetric Boolean Functions (RSBFs) by Kavut, Maitra and Yucel. In this paper, we present several 9-variable Boolean functions having nonlinearity of 242, which we obtain by suitably generalizing the classes of RSBFs and Dihedral Symmetric Boolean Functions (DSBFs). These functions do not have any zero in the Walsh spectrum values, hence they cannot be made balanced easily. This result also shows that the covering radius of the first order Reed-Muller code R(1, 9) is at least 242.

Rotation symmetric functions are a variant of symmetric functions, such that function values are unchanged by any rotation of digits in input vectors. Since rotation symmetric functions are resistant to linear attacks due to their nonlinear characteristics, they are promising for cryptographic applications. However, there are few studies on how to represent them. This paper focuses on representation using functional decomposition and decision diagrams, as a steppingstone to find a compact representation of rotation symmetric functions. Experimental results show that the presented representation is promising.

The number and probability of canalizing functions