Balanced Rotation Symmetric Boolean Functions Research Articles

Balanced Even-Variable Rotation Symmetric Boolean Functions with Optimal Algebraic Immunity, Maximum Algebraic Degree and Higher Nonlinearity

Rotation symmetric Boolean functions are good candidates for stream ciphers because they have such advantages as simple structure, high operational speed and low implement cost. Recently, Mesnager et al. proposed for the first time an efficient method to construct balanced rotation symmetric Boolean functions with optimal algebraic immunity and good nonlinearity for an arbitrary even number of variables. However, the algebraic degree of their constructed [Formula: see text]-variable ([Formula: see text]) function is always less than the maximum value [Formula: see text]. In this paper, by modifying the support of Boolean functions from Mesnager et al.’s construction, we present two new constructions of balanced even-variable rotation symmetric Boolean functions with optimal algebraic immunity as well as higher algebraic degree and nonlinearity. The algebraic degree of Boolean functions in the first construction reaches the maximum value [Formula: see text] if [Formula: see text] is odd and [Formula: see text] or [Formula: see text] for integer [Formula: see text], while that of the second construction reaches the maximum value for all [Formula: see text]. Moreover, the nonlinearities of Boolean functions in both two constructions are higher than that of Mesnager et al.’s construction.

Balanced Rotation Symmetric Boolean Functions With Good Autocorrelation 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.

A construction method of balanced rotation symmetric Boolean functions on arbitrary even number of variables with optimal algebraic immunity

Rotation symmetric Boolean functions incorporate a super-class of symmetric functions which represent an attractive corpus for computer investigation. These functions have been investigated from the viewpoints of bentness and correlation immunity and have also played a role in the study of nonlinearity. In the literature, many constructions of balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity have been derived. While it seems that the construction of balanced even-variable rotation symmetric Boolean functions with optimal algebraic immunity is very hard work to breakthrough. In this paper, we present for the first time a construction of balanced rotation symmetric Boolean functions on an arbitrary even number of variables with optimal algebraic immunity by modifying the support of the majority function. The nonlinearity of the newly constructed rotation symmetric Boolean functions is also derived.

Balanced $$2^k$$ 2 k -variable rotation symmetric Boolean functions with optimal algebraic immunity

Rotation symmetric Boolean functions have been extensive studied because of their importance in cryptography. These functions are invariant under circular translation of indices. In this paper, we propose a new construction of \(2^k\)-variable balanced rotation symmetric Boolean functions with optimal algebraic immunity. The nonlinearity of our new functions is significantly higher than all previously obtained even-variable rotation symmetric Boolean functions with optimal algebraic immunity.

Constructions of balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity

Rotation symmetric Boolean functions have been used as components of different cryptosystems. In this paper, two classes of balanced rotation symmetric Boolean functions having optimal algebraic immunity on odd number of variables are constructed. We give a lower bound on the algebraic degree of the first class of functions, and prove that the n-variable functions in the second class has optimal algebraic degree if n≠2m+1 for m>2. Moreover, it is shown that both classes of functions have much better nonlinearity than all the previously obtained rotation symmetric Boolean functions with optimal algebraic immunity, and have good behavior against fast algebraic attacks at least for small numbers of input variables.

Construction of RSBFs with improved cryptographic properties to resist differential fault attack on grain family of stream ciphers

In recent literature, the differential fault analysis (DFA) on Grain family of stream ciphers has been shown to exploit the low algebraic degree of the derivative of the nonlinear combining function h of the stream cipher, h(x) ? h(x ? ?). The low algebraic degree allows the DFA adversary to create a linearly independent system of equations generated from the faulty and fault-free keystreams and use these equations to extract the initial state of the NFSR and LFSR stages in the stream cipher. In this paper, we propose a construction scheme for rotation symmetric Boolean functions (RSBFs) h(x) along with an orbit-tuple flip based iterative hill-climbing based construction algorithm for balanced RSBFs with high nonlinearity, low absolute indicator value of global avalanche characteristics (GAC), and high algebraic degree of h(x) ? h(x ? ?). The construction algorithm is scalable for higher input variables like n = 9,10,11 as shown in the paper. We find some interesting autocorrelation spectra and Walsh spectra properties for the class of RSBFs and then use them in the construction of RSBFs with improved cryptographic properties. We present the cryptographic properties of the RSBFs constructed for high input variables which can be used to make DFA attack harder using the existing techniques.

Construction of balanced rotation symmetric Boolean functions with optimal algebraic immunity

Algebraic immunity is a new cryptographic criterion proposed against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in many stream ciphers should possess high algebraic immunity. This paper presents one main result to find balanced rotation symmetric Boolean functions with maximum algebraic immunity. Through swapping the values of two orbits of rotation class of the majority function, a class of 4k+1 variable Boolean functions with maximum algebraic immunity is constructed. The function f(x) we construct always has terms of degree n-2 independence of what ever n is. And the nonlinearity of f(x) is relatively good for large n.

Counting rotation symmetric functions using Polya’s theorem

Homogeneous rotation symmetric (invariant under cyclic permutation of the variables) Boolean functions have been extensively studied in recent years due to their applications in cryptography. In this paper we give an explicit formula for the number of homogeneous rotation symmetric functions over the finite field GF(pm) using Polya’s enumeration theorem, which completely solves the open problem proposed by Yuan Li in 2008. This result simplifies the proof and the nonexplicit counting formula given by Shaojing Fu et al. over the field GF(p). This paper also gives an explicit count for n-variable balanced rotation symmetric Boolean functions with n=pq, where p and q are distinct primes. Previous work only gave an explicit count for the case where n is prime and lower bounds for the case where n is a prime power.

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

In designing cryptographic Boolean functions, it is challenging to achieve at the same time the desirable features of algebraic immunity, balancedness, nonlinearity, and algebraic degree for necessary resistance against algebraic attack, correlation attack, Berlekamp–Massey attack, etc. This paper constructs balanced rotation symmetric Boolean functions on n variables where n=2p and p is an odd prime. We prove the construction has an optimal algebraic immunity and is of high nonlinearity. We check that, at least for those primes p which are not of the form of a power of two plus one, the algebraic degree of the construction achieves in fact the upper bound n−1.

Some results on the algebraic immunity of Boolean functions

From the motivation of algebraic attacks on stream and block ciphers, the concept of algebraic immunity (AI) of a Boolean function was introduced and studied extensively. High algebraic immunity is a necessary condition for resisting algebraic attacks. In this paper, we give some lower bounds on the algebraic immunity of Boolean functions. The results are applied to give lower bounds on the AI of symmetric Boolean functions and rotation symmetric Boolean functions. Some balanced rotation symmetric Boolean functions with their AI near the maximum possible value ( n/2 (are constructed.

Balanced rotation symmetric boolean functions with maximum algebraic immunity

Rotation symmetric Boolean functions (RSBFs) that are invariant under circular translation of indices have been used as components of different cryptosystems. In this paper, even-variable-balanced RSBFs with maximum algebraic immunity (AI) are investigated. At first, we give an original construction of 2m-variable-balanced RSBFs with maximum AI. Then we improve the construction to obtain more 2m-variable-balanced RSBFs with maximum AI, and these new RSBFs have higher non-linearity than all previously obtained RSBFs. Further, we generalise our construction of 2m-variable RSBFs to a new construction that can generate any even-variable RSBFs.