Boolean Function Representation of S-Boxes and Boolean Permutations

This paper proposes a new recursive method for constructing (n+1)-bit Boolean permutations from n-bit ones, and proves that all (n+1)-bit Boolean permutations can be constructed by the method, thereby solving enumeration problem of the kind of Boolean permutations. Furthermore, Boolean permutations with good cryptographic properties can be constructed by adding some restrictive conditions to the method.

S-boxes, typically the only nonlinear part of a block cipher, are the heart of symmetric cryptographic primitives. They significantly impact the cryptographic strength and the implementation characteristics of an algorithm. Due to their simplicity, quadratic vectorial Boolean functions are preferred when efficient implementations for a variety of applications are of concern. Many characteristics of a function stay invariant under affine equivalence. So far, all 6-bit Boolean functions, 3- and 4-bit permutations have been classified up to affine equivalence. At FSE 2017, Bozoliv et al. presented the first classification of 5-bit quadratic permutations. In this work, we propose an adaptation of their work resulting in a highly efficient algorithm to classify n x m functions for n ≥ m. Our algorithm enables for the first time a complete classification of 6-bit quadratic permutations as well as all balanced quadratic functions for n ≤ 6. These functions can be valuable for new cryptographic algorithm designs with efficient multi-party computation or side-channel analysis resistance as goal. In addition, we provide a second tool for finding decompositions of length two. We demonstrate its use by decomposing existing higher degree S-boxes and constructing new S-boxes with good cryptographic and implementation properties.

Internet of things need the encryption algorithm with a piece of small area with small-scale. This paper obtains some rotation Boolean permutations by the matrix of linear expressions, and constructs three methods of rotation nonlinear Boolean permutations. The sub-functions of the three permutations have some properties with three monomials, high degree and 2-algebra immunity. Finally, we derive the disjoint spectra Boolean functions.

Boolean functions are used as nonlinear filter functions and combiner functions in several stream ciphers. The security of these stream ciphers largely depends upon cryptographic properties of Boolean functions. Finding a balanced Boolean function with optimal cryptographic properties is an open research problem in the cryptographic community. Since the number of n-variable Boolean functions is $$2^{2^n}$$ , it is not computationally feasible to search the entire space of such functions for cryptographically significant functions when $$n \ge 6$$ . In general, the construction of Boolean functions with optimal or near optimal cryptographically significant properties is formulated as combinatorial optimization problems. In this paper, we apply the Genetic algorithm with the integration of a local search procedure called hybrid GA (HGA) searching for the Boolean functions with high nonlinearity and low autocorrelation. The performance of the Genetic algorithm depends upon the tuning parameters such as crossover mechanism, mutation probability, selection criteria, and the choice of fitness/cost function. To achieve an optimal trade-off among two cryptographic properties, the choice of the cost function plays an important role for finding an optimal solution. So our main focus is to construct balanced Boolean functions using HGA with a new cost function and compare it with existing cost functions. Our experimental results shows that the HGA is more efficient than GA and the new cost function is more efficient than existing cost functions for reaching the optimal solutions.

Cryptographic applications of Boolean functions are meant to have some cryptographic properties, those properties are built to thwart cryptanalysis of certain kinds, and multiple cryptographic properties are usually required for a Boolean function to be used in cryptographic algorithm design, expected to resist some known attacks to the cryptographic algorithms. Therefore, the primary applications of cryptographic Boolean functions are the design of cryptographic algorithms, particularly stream cipher and block cipher algorithms. This chapter will discuss some applications of Boolean functions with some cryptographic properties in the areas beyond cryptographic algorithm design, where the involved Boolean functions are primary building blocks.

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

The cryptographic primitives such as Boolean function and S-Box are used as a building block to design stream cipher and block cipher, respectively. However, it is important to evaluate the cryptographic strength to measure the security of such crytposystem resistance against important cryptanalytic attacks. We develop a new tool called Boolean function and S-Box analysis tool (BSAT) to evaluate the cryptographic strength of the Boolean function and S-Box. As the size of S-Box increases, it takes a lot of computational effort to evaluate all the required cryptographic properties. In this paper, we develop a tool to evaluate important cryptographic properties of both Boolean function and S-Box to minimize the computational time. The computational time of BSAT is lesser than the SET tool in terms of calculating the algebraic normal form and algebraic degree. Our BSAT takes only 0.404 s to evaluate the AES properties faster than the SET, which takes 650 millisecond (0.65 s).

This paper investigates the optimum OBDD representation problem based on two classes of Boolean functions. The first class is defined by OBDDs, in which the number of non-terminal nodes is equal to the number of input variables. We refer to such OBDDs and their corresponding Boolean functions as thin OBDDs and thin Boolean functions. The second class is the thin factored Boolean functions, which is defined by factored forms with each variable appearing exactly once. Many interesting properties of these two classes of functions are presented. Based on which, a revised dynamic shortest cube first OBDD variable ordering algorithm is developed. This algorithm is shown to be optimum for thin factored Boolean functions.

The number and probability of canalizing functions

Boolean function is one of the most fundamental computation models in theoretical computer science. The two most common representations of Boolean functions are Fourier transform and real polynomial form. Applying analytic tools under these representations to the study Boolean functions has led to fruitful research in many areas such as complexity theory, learning theory, inapproximability, pseudorandomness, metric embedding, property testing, threshold phenomena, social choice, etc. In this thesis, we focus on \emph{sparse representations} of Boolean function in both Fourier transform and polynomial form, and obtain the following new results. A classical result of Rothschild and van Lint asserts that if every non-zero Fourier coefficient of a Boolean function $f$ over $\mathbb{F}_2^{n}$ has the same absolute value, namely $|\hat{f}(\alpha)|=1/2^k$ for every $\alpha$ in the Fourier support of $f$, then $f$ must be the indicator function of some affine subspace of dimension $n-k$. Here we slightly generalize their result, and show that Boolean functions whose Fourier coefficients take values in the set $\{-2/2^k, -1/2^k, 0, 1/2^k, 2/2^k\}$ are indicator functions of two disjoint affine subspaces of dimension $n-k$ or four disjoint affine subspaces of dimension $n-k-1$. Our main technical tools are results from additive combinatorics which offer tight bounds on the affine span size of a subset of $\mathbb{F}_2^{n}$ when the doubling constant of the subset is small. For polynomial representation of Boolean function, we study the distribution of the number of non-zero coefficients of \emph{random} Boolean functions. For a random Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$, i.e., a function whose value at each point on the Boolean cube is chosen independently and uniformly at random from $\{0,1\}$, in real polynomial representation, we give several bounds and concentration results about the distribution of the sparsity of $f$.

Using the derivative of the Boolean function and the e-derivative defined by ourselves as research tools, we go deep into the internal structure of the Boolean function values. Cryptographic properties such as algebraic immunity, correlation immune and algebraic degree of H Boolean functions with Hamming weight of with diffusibility and the relationship between these properties are studied. Then we get the results of the mathematical expression of linear annihilators, the values of algebraic degree and correlation immune order, and so on.

Since 1970, Boolean functions have been the focus of a lot of attention in cryptography. An important topic in symmetric ciphers concerns the cryptographic properties of Boolean functions and constructions of Boolean functions with good cryptographic properties, that is, good resistance to known attacks. An important progress in cryptanalysis areas made in 2003 was the introduction by Courtois and Meier of algebraic attacks and fast algebraic attacks which are very powerful analysis concepts and can be applied to almost all cryptographic algorithms. To study the resistance against algebraic attacks, the notion of algebraic immunity has been introduced. In this paper, we use a parameter introduced by Liu and al., called fast algebraic immunity, as a tool to measure the resistance of a cryptosystem (involving Boolean functions) to fast algebraic attacks. We prove an upper bound on the fast algebraic immunity. Using our upper bound, we establish the weakness of trace inverse functions against fast algebraic attacks confirming a recent result of Feng and Gong.

In this paper, the analysis of cyclotomic coset has been done for construction of Boolean functions. Boolean function used in cryptographic systems should have good cryptographic properties against standard or fast algebraic attacks. The cyclotomic coset plays very important role in cryptography. By using cyclotomic coset leader and minimal polynomial, we construct more balanced Boolean function. In this paper, cryptographic properties; balancedness, algebraic-degree, algebraic-immunity and non-linearity are also defined in preliminaries.

International audience

A further characterization of the bent-negabent functions is presented. Based on the concept of complete mapping polynomial, we provide a necessary and sufficient condition for a class of quadratic Boolean functions to be bent-negabent. A new characterization of negabent functions can be described by using the parity of Hamming weight. We further generalize the classical convolution theorem and give the nega-Hadamard transform of the composition of a Boolean function and a vectorial Boolean function. The nega-Hadamard transform of a generalized indirect sum is calculated by this composition method.