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.

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.

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.

Constructing APN or 4-differentially uniform permutations achieving all the necessary criteria is an open problem, and the research on it progresses slowly. In ACISP 2011, Carlet put forth an idea for constructing differentially uniform permutations using extension fields, which was illustrated with a construction of a 4-differentially uniform (n,n)-permutation. The permutation has optimum algebraic degree and very good nonlinearity. However, it was proved to be a permutation only for n odd. In this note, we investigate further the construction of differentially uniform permutations using extension fields, and construct a 4-differentially uniform (n,n)-permutation for any n. These permutations also have optimum algebraic degree and very good nonlinearity. Moreover, we consider a more general type of construction, and illustrate it with an example of a 4-differentially uniform (n,n)-permutation with good cryptographic properties.

In algebraic attack on stream ciphers based on LFSRs, the secret key is found by solving an overdefined system of multivariate equations. There are many known algorithms from different point of view to solve the problem, such as linearization, relinearization, XL and Grobner Basis. The simplest method, linearization, treats each monomial of different degrees as a new variable, and consists of variables (the degree of the system of equations is denoted by d). Thus it needs at least equations, i.e. keystream bits to recover the secret key by Gaussian reduction or other. In this paper we firstly propose a concept, called equivalence of LFSRs. On the basis of it, we present a constructive method that can solve an overdefined system of multivariate equations with less keystream bits by extending the primitive polynomial.

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.