Construction of multiple output functions is one of the most important problems in the design and analysis of stream ciphers. Generally, such a function has to be satisfied with several criteria, such as high nonlinearity, resiliency and high algebraic degree. But there are mutual restraints among the cryptographic parameters. Finding a way to achieve the optimization is always regarded as a hard task. In this paper, by using the disjoint linear codes and disjoint spectral functions, two classes of resilient multiple output functions are obtained. It has been proved that the obtained functions have high nonlinearity and high algebraic degree.

Semi-bent functions have important applications in cryptography and coding theory. 2-rotation symmetric semi-bent functions are a class of semi-bent functions with the simplicity for efficient computation because of their invariance under 2-cyclic shift. However, no construction of 2-rotation symmetric semi-bent functions with algebraic degree bigger than 2 has been presented in the literature. In this paper, we introduce four classes of 2m-variable 2-rotation symmetric semi-bent functions including balanced ones. Two classes of 2-rotation symmetric semi-bent functions have algebraic degree from 3 to m for odd m≥3, and the other two classes have algebraic degree from 3 to m/2 for even m≥6 with m/2 being odd.

Boolean functions and vectorial Boolean functions are the most important components of stream ciphers. Their cryptographic properties are crucial to the security of the underlying ciphers. And how to construct such functions with good cryptographic properties is a nice problem that worth to be investigated. In this paper, using two small nonlinear functions with t-1 resiliency, we provide a method on constructing t-resilient n variables Boolean functions with strictly almost optimal nonlinearity >2n-1-2n/2 and optimal algebraic degree n-t-1. Based on the method, we give another construction so that a large class of resilient vectorial Boolean functions can be obtained. It is shown that the vectorial Boolean functions also have strictly almost optimal nonlinearity and optimal algebraic degree.

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.

Semi-bent functions have very high nonlinearity and hence they have many applications in symmetric-key cryptography, binary sequence design for communications, and combinatorics. In this paper, we focus on studying the additive autocorrelation of semi-bent functions. We provide a lower bound on the maximum additive autocorrelation absolute value of semi-bent functions with three-level additive autocorrelation. Semi-bent functions with three-level additive autocorrelation achieving this bound with equality are said to have perfect three-level additive autocorrelation. We present two classes of balanced semi-bent functions with optimal algebraic degree and perfect three-level additive autocorrelation.

Research on permutation polynomials over the finite field F22k with significant cryptographical properties such as possibly low differential uniformity, possibly high nonlinearity and algebraic degree has attracted a lot of attention and made considerable progress in recent years. Once used as the substitution boxes (S-boxes) in the block ciphers with Substitution Permutation Network (SPN) structure, this kind of polynomials can have a good performance against the classical cryptographic analysis such as linear attacks, differential attacks and the higher order differential attacks. In this paper we put forward a new construction of differentially 4-uniformity permutations over F22k by modifying the inverse function on some specific subsets of the finite field. Compared with the previous similar works, there are several advantages of our new construction. One is that it can provide a very large number of Carlet-Charpin-Zinoviev equivalent classes of functions (increasing exponentially). Another advantage is that all the functions are explicitly constructed, and the polynomial forms are obtained for three subclasses. The third advantage is that the chosen subsets are very large, hence all the new functions are not close to the inverse function. Therefore, our construction may provide more choices for designing of S-boxes. Moreover, it has been checked by a software programm for k=3 that except for one special function, all the other functions in our construction are Carlet-Charpin-Zinoviev equivalent to the existing ones.

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.

T-function is a kind of cryptographic function which is shown to be useful in various applications. It is known that any function f on F2n or Z2n automatically deduces a unique polynomial fF ∈ F2n[x] with degree ≤ 2n-1. In this letter, we study an algebraic property of fF while f is a T-function. We prove that for a single cycle T-function f on F2n or Z2n, deg fF=2n-2 which is optimal for a permutation. We also consider a kind of widely used T-function in many cryptographic algorithms, namely the modular addition function Ab(x)=x+b ∈ Z2n[x]. We demonstrate how to calculate deg Ab F from the constant value b. These results can facilitate us to evaluate the immunity of the T-function based cryptosystem against some known attacks such as interpolation attack and integral attack.

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 study, we construct balanced Boolean functions with a high nonlinearity and an optimum algebraic degree for both odd and even dimensions. Our approach is based on modifying functions from the Maiorana-McFarland's superclass, which has been introduced by Carlet. A drawback of Maiorana-McFarland's function is that their restrictions obtained by fixing some variables in their input are affine. Affine functions are cryptographically weak functions, so there is a risk that this property will be exploited in attacks. Due to the contribution of Carlet, our constructions do not have the potential weakness that is shared by the Maiorana-McFarland construction or its modifications.