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.

In recent years, algebraic attacks and fast algebraic attacks have received a lot of attention in the cryptographic community. There are three Boolean functions achieving optimal algebraic immunity based on primitive element of F2n. The support of Boolean functions in [1]-[3] have the same parameter s, which makes us have a large number of Boolean functions with good properties. However, we prove that the Boolean functions are affine equivalence when s takes different values.

We propose a recursive algorithm to reduce the computational complexity of the r-order nonlinearity of n-variable Boolean functions. Applying the algorithm and using the sufficient and necessary condition put forward by [1] to cut the vast majority of useless search branches, we show that the covering radius of the Reed-Muller Code R(3, 7) in R(5, 7) is 20.

It is known that correlation-immune (CI) Boolean functions used in the framework of side channel attacks need to have low Hamming weights. In this letter, we determine all unknown values of the minimum Hamming weights of d-CI Boolean functions in n variables, for d ≤ 5 and n ≤ 13.

Boolean functions used in stream ciphers and block ciphers should have high second-order nonlinearity to resist several known attacks and some potential attacks which may exist but are not yet efficient and might be improved in the future. The second-order nonlinearity of Boolean functions also plays an important role in coding theory, since its maximal value equals the covering radius of the second-order Reed-Muller code. But it is an extremely hard task to calculate and even to bound the second-order nonlinearity of Boolean functions. In this paper, we present a lower bound on the second-order nonlinearity of the generalized Maiorana-McFarland Boolean functions. As applications of our bound, we provide more simpler and direct proofs for two known lower bounds on the second-order nonlinearity of functions in the class of Maiorana-McFarland bent functions. We also derive a lower bound on the second-order nonlinearity of the functions which were conjectured bent by Canteaut and whose bentness was proved by Leander, by further employing our bound.

This paper improves the family size of quadrature amplitude modulation (QAM) complementary sequences with binary inputs. By employing new mathematical description: B-type-2 of 4q-QAM constellation (integer q ≥ 2), a new construction yielding 4q-QAM complementary sequences (CSs) with length 2m (integer m ≥ 2) is developed. The resultant sequences include the known QAM CSs with binary inputs as special cases, and the family sizes of new sequences are approximately 22·2q-4q-1(22·2q-3-1) times as many as the known. Also, both new sequences and the known have the same the peak envelope power (PEP) upper bounds, when they are used in an orthogonal frequency-division multiplexing communication system.

We present an explicit construction of a MAJn-2 °MAJn-2 circuit computing MAJn for every odd n≥7. This gives a partial solution to an open problem by Kulikov and Podolskii (Proc. of STACS 2017, Article No.49).

As a with-carry analog (based on modular arithmetic) of the usual Walsh-Hadamard transform (WHT), arithmetic Walsh transform (AWT) has been used to obtain analogs of some properties of Boolean functions which are important in the design and analysis of cryptosystems. The existence of nonzero linear structure of Boolean functions is an important criterion to measure the weakness of these functions in their cryptographic applications. In this paper, we find more analogs of linear structures of Boolean functions from AWT. For some classes of n-variable Boolean functions f, we find necessary and sufficient conditions for the existence of an invariant linear structure and a complementary linear structure 1n of f. We abstract out a sectionally linear relationship between AWT and WHT of n-variable balanced Boolean functions f with linear structure 1n. This result show that AWT can characterize cryptographic properties of these functions as long as WHT can. In addition, for a diagonal Boolean function f, a recent result by Carlet and Klapper says that the AWT of f can be expressed in terms of the AWT of a diagonal Boolean function of algebraic degree at most 3 in a larger number of variables. We provide for the result a complete and more modular proof which works for both even and odd weights (of the parameter c in the Corollary 19 by Carlet and Klapper (DCC 73(2): 299-318, 2014).

Recent research has shown that the class of rotation symmetric Boolean functions is beneficial to cryptographics. In this paper, for an odd prime p, two sufficient conditions for p-variable rotation symmetric Boolean functions to be 1-resilient are obtained, and then several concrete constructions satisfying the conditions are presented. This is the first time that resilient rotation symmetric Boolean functions have been systematically constructed. In particular, we construct a class of 2-resilient rotation symmetric Boolean functions when p=2m+1 for m ≥ 4. Moreover, several classes of 1-order correlation immune rotation symmetric Boolean functions are also got.

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.

By investigating the properties that the offsets should satisfy, this letter presents a brief proof of general QAM Golay complementary sequences (GCSs) in Cases I-III constructions. Our aim is to provide a brief, clear, and intelligible derivation so that it is easy for the reader to understand the known Cases I-III constructions of general QAM GCSs.

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 explore possibilities and difficulties to prove super-quadratic formula size lower bounds from the following aspects. First, we consider recursive Boolean functions and prove their general formula size upper bounds. We also discuss recursive Boolean functions based on exact 2-bit functions. We show that their formula complexity are at least Ω(n2). Hence they can be candidate Boolean functions to prove super-quadratic formula size lower bounds. Next, we consider the reason of the difficulty of resolving the formula complexity of the majority function in contrast with the parity function. In particular, we discuss the structure of an optimal protocol partition for the Karchmer-Wigderson communication game.

Based on the non-standard generalized Boolean functions (GBFs) over Z4, we propose a new method to convert those functions into the 16-QAM Golay complementary sequences (CSs). The resultant 16-QAM Golay CSs have the upper bound of peak-to-mean envelope power ratio (PMEPR) as low as 2. In addition, we obtain multiple 16-QAM Golay CSs for a given quadrature phase shift keying (QPSK) Golay CS.

In this paper, we provide several methods to construct nonlinear resilient functions with multiple good cryptographic properties, including high nonlinearity, high algebraic degree, and non-existence of linear structures. Firstly, we present an improvement on a known construction of resilient S-boxes such that the nonlinearity and the algebraic degree will become higher in some cases. Then a construction of highly nonlinear t-resilient Boolean functions without linear structures is given, whose algebraic degree achieves n-t-1, which is optimal for n-variable t-resilient Boolean functions. Furthermore, we construct a class of resilient S-boxes without linear structures, which possesses the highest nonlinearity and algebraic degree among all currently known constructions.

In this paper, we present an average-case efficient algorithm to resolve the problem of determining whether two Boolean functions in trace representation are identical. Firstly, we introduce a necessary and sufficient condition for null Boolean functions in trace representation, which can be viewed as a generalization of the well-known additive Hilbert-90 theorem. Based on this condition, we propose an algorithmic method with preprocessing to address the original problem. The worst-case complexity of the algorithm is still exponential; its average-case performance, however, can be improved. We prove that the expected complexity of the refined procedure is O(n), if the coefficients of input functions are chosen i.i.d. according to the uniform distribution over F2n; therefore, it performs well in practice.

Two classes of 3rd order correlation immune symmetric Boolean functions have been constructed respectively in [1] and [2], in which some interesting phenomena of the algebraic degree have been observed as well. However, a good explanation has not been given. In this paper, we obtain the formulas for the degree of these functions, which can well explain the behavior of their degree.

In this paper, we give some results towards the conjecture that σ2t+1l-1,2t are the only nonlinear balanced elementary symmetric Boolean functions where t and l are positive integers. At first, a unified and simple proof of some earlier results is shown. Then a property of balanced elementary symmetric Boolean functions is presented. With this property, we prove that the conjecture is true for n=2m+2t-1 where m,t (m>t) are two non-negative integers, which verified the conjecture for a large infinite class of integer n.