In 2015, Carlet and Tang [Des. Codes Cryptogr. 76(3): 571-587, 2015] proposed a concept called enhanced Boolean functions and a class of such kind of functions on odd number of variables was constructed. They proved that the constructed functions in this class have optimal algebraic immunity if the numbers of variables are a power of 2 plus 1 and at least sub-optimal algebraic immunity otherwise. In addition, an open problem that if there are enhanced Boolean functions with optimal algebraic immunity and maximal algebraic degree n-1 on odd variables n≠2k+1 was proposed. In this letter, we give a negative answer to the open problem, that is, we prove that there is no enhanced Boolean function on odd n≠2k+1 variables with optimal algebraic immunity and maximal algebraic degree n-1.

Boolean functions used in the filter model of stream ciphers should have balancedness, large nonlinearity, optimal algebraic immunity and high algebraic degree. Besides, one more criterion called strict avalanche criterion (SAC) can be also considered. During the last fifteen years, much work has been done to construct balanced Boolean functions with optimal algebraic immunity. However, none of them has the SAC property. In this paper, we first present a construction of balanced Boolean functions with SAC property by a slight modification of a known method for constructing Boolean functions with SAC property and consider the cryptographic properties of the constructed functions. Then we propose an infinite class of balanced functions with optimal algebraic immunity and SAC property in odd number of variables. This is the first time that such kind of functions have been constructed. The algebraic degree and nonlinearity of the functions in this class are also determined.

Rotation symmetric Boolean functions which are invariant under the action of cyclic group have been used in many different cryptosystems. This paper presents a new construction of balanced odd-variable rotation symmetric Boolean functions with optimum algebraic immunity. It is checked that, at least for some small variables, such functions have very good behavior against fast algebraic attacks. Compared with some known rotation symmetric Boolean functions with optimum algebraic immunity, the new construction has really better nonlinearity. Further, the algebraic degree of the constructed functions is also high enough.

To resist algebraic and fast algebraic attacks, Boolean functions used in stream ciphers should have optimal algebraic immunity and good fast algebraic immunity. One challenge of cryptographic Boolean functions is to determine their ability to resist fast algebraic attacks, which can be measured by their fast algebraic immunities. In this letter, we determine the exact values of fast algebraic immunity of the majority function of 2m and 2m+1 variables. This is the first time that the exact values of the fast algebraic immunity of an infinite class of symmetric Boolean functions with optimal algebraic immunity are determined.

Since 2008, three different classes of Boolean functions with optimal algebraic immunity have been proposed by Carlet and Feng [2], Wang et al.[8] and Chen et al.[3]. We call them C-F functions, W-P-K-X functions and C-T-Q functions for short. In this paper, we propose three affine equivalent classes of Boolean functions containing C-F functions, W-P-K-X functions and C-T-Q functions as a subclass, respectively. Based on the affine equivalence relation, we construct more classes of Boolean functions with optimal algebraic immunity. Moreover, we deduce a new lower bound on the nonlinearity of C-F functions, which is better than all the known ones.

Rotation symmetric Boolean functions (RSBFs) that are invariant under circular translation of indices have been used as components of different cryptosystems. In this paper, odd-variable balanced RSBFs with maximum algebraic immunity (AI) are investigated. We provide a construction of n-variable (n=2k+1 odd and n ≥ 13) RSBFs with maximum AI and nonlinearity ≥ 2n-1-¥binom{n-1}{k}+2k+2k-2-k, which have nonlinearities significantly higher than the previous nonlinearity of RSBFs with maximum AI.

The global avalanche characteristics measure the overall avalanche properties of Boolean functions, an n-variable balanced Boolean function of the sum-of-square indicator reaching σƒ=22n+2n+3 is an open problem. In this paper, we prove that there does not exist a balanced Boolean function with σƒ=22n+2n+3 for n≥4, if the hamming weight of one decomposition function belongs to the interval Q*. Some upper bounds on the order of propagation criterion of balanced Boolean functions with n (3≤n≤100) variables are given, if the number of vectors of propagation criterion is equal and less than 7·2n-3-1. Two lower bounds on the sum-of-square indicator for balanced Boolean functions with optimal autocorrelation distribution are obtained. Furthermore, the relationship between the sum-of-squares indicator and nonlinearity of balanced Boolean functions is deduced, the new nonlinearity improves the previously known nonlinearity.

In this paper, we put forward a new method to construct n-variable Boolean functions with optimal algebraic immunity based on the factorization of n. Computer investigations for small values of n indicate that a class of Boolean functions constructed by the new method has a very good nonlinearity and also a good behavior against fast algebraic attacks.

Because of the algebraic attacks, a high algebraic immunity is now an important criteria for Boolean functions used in stream ciphers. In 2011, X.Y. Zeng et al. proposed three constructions of balanced Boolean functions with maximum algebraic immunity, the constructions are based on univariate polynomial representation of Boolean functions. In this paper, we will improve X.Y. Zeng et al.' constructions to obtain more even-variable Boolean functions with maximum algebraic immunity. It is checked that, our new functions can have as high nonlinearity as X.Y. Zeng et al.' functions.

In this comment, an inequality of algebraic immunity of the sum of two Boolean functions is pointed out to be generally incorrect. Then we present some results on how to impose conditions such that the inequality is true. Finally, complete proofs of two existing results are given.

It is well known that Boolean functions used in stream and block ciphers should have high algebraic immunity to resist algebraic attacks. Up to now, there have been many constructions of Boolean functions achieving the maximum algebraic immunity. In this paper, we present several constructions of rotation symmetric Boolean functions with maximum algebraic immunity on an odd number of variables which are not symmetric, via a study of invertible cyclic matrices over the binary field. In particular, we generalize the existing results and introduce a new method to construct all the rotation symmetric Boolean functions that differ from the majority function on two orbits. Moreover, we prove that their nonlinearities are upper bounded by .

Based on Tu-Deng's conjecture and the Tu-Deng function, in 2010, X. Tang et al. proposed a class of Boolean functions in even variables with optimal algebraic degree, very high nonlinearity and optimal algebraic immunity. In this corresponding, we consider the concatenation of Tang's function and another Boolean function, and study its cryptographic properties. With this idea, we propose a class of 1-resilient Boolean functions in odd variables with optimal algebraic degree, good nonlinearity and suboptimal algebraic immunity based on Tu-Deng's conjecture.

In this note, we go further on the “basis exchange” idea presented in [2] by using Mobious inversion. We show that the matrix S1(f)S0(f)-1 has a nice form when f is chosen to be the majority function, where S1(f) is the matrix with row vectors υk(α) for all α ∈ 1f and S0(f)=S1(f ⊕ 1). And an exact counting for Boolean functions with maximum algebraic immunity by exchanging one point in on-set with one point in off-set of the majority function is given. Furthermore, we present a necessary condition according to weight distribution for Boolean functions to achieve algebraic immunity not less than a given number.

In this paper, we deal with the algebraic immunity of the symmetric Boolean functions. The algebraic immunity is a property which measures the resistance against the algebraic attacks on symmetric ciphers. It is well known that the algebraic immunity of the symmetric Boolean functions is completely determined by a narrow class of annihilators with low degree which is denoted by G(n,). We study and determine the weight support of part of these functions. Basing on this, we obtain some relations between the algebraic immunity of a symmetric Boolean function and its simplified value vector. For applications, we put forward an upper bound on the number of the symmetric Boolean functions with algebraic immunity at least d and prove that the algebraic immunity of the symmetric palindromic functions is not high.

In this paper, we constructed six infinite classes of balanced Boolean functions. These six classes of Boolean functions achieved optimal algebraic degree, optimal algebraic immunity and high nonlinearity. Furthermore, we gave the proof of the lower bound of the nonlinearities of these balanced Boolean functions and proved the better lower bound of nonlinearity for Carlet-Feng's Boolean function.

In this paper, we explicitly construct a large class of symmetric Boolean functions on 2k variables with algebraic immunity not less than d, where integer k is given arbitrarily and d is a given suffix of k in binary representation. If let d = k, our constructed functions achieve the maximum algebraic immunity. Remarkably, 2⌊ log2k ⌋ + 2 symmetric Boolean functions on 2k variables with maximum algebraic immunity are constructed, which are much more than the previous constructions. Based on our construction, a lower bound of symmetric Boolean functions with algebraic immunity not less than d is derived, which is 2⌊ log2d ⌋ + 2(k-d+1). As far as we know, this is the first lower bound of this kind.

A method to construct Boolean functions with maximum algebraic immunity have been proposed in . Based on that method, we propose a different method to construct Boolean functions on even variables with maximum algebraic immunity in this letter. By counting on our construction, a lower bound of the number of such Boolean functions is derived, which is the best among all the existing lower bounds.

It is known that Boolean functions used in stream ciphers should have good cryptographic properties to resist fast algebraic attacks. In this paper, we study a new class of Boolean functions with good cryptographic properties: balancedness, optimum algebraic degree, optimum algebraic immunity and a high nonlinearity.