Symmetric Boolean Functions Research Articles

A divisibility approach to the open boundary cases of Cusick-Li-Stǎnicǎ’s conjecture

In this paper we compute the exact 2-divisibility of exponential sums associated to elementary symmetric Boolean functions. Our computation gives an affirmative answer to most of the open boundary cases of Cusick-Li-Sta?nica?'s conjecture. As a byproduct, we prove that the 2-divisibility of these families satisfies a linear recurrence. In particular, we provide a new elementary method to compute 2-divisibility of symmetric Boolean functions.

Dual lower bounds for approximate degree and Markov–Bernstein inequalities

The ε-approximate degree of a Boolean function f:{−1,1}n→{−1,1} is the minimum degree of a real polynomial that approximates f to within error ε in the ℓ∞ norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program. Our first result resolves the ε-approximate degree of the two-level AND–OR tree for any constant ε>0. We show that this quantity is Θ(n), closing a line of incrementally larger lower bounds. The same lower bound was recently obtained independently by Sherstov (Theory Comput. 2013) using related techniques. Our second result gives an explicit dual polynomial that witnesses a tight lower bound for the approximate degree of any symmetric Boolean function, addressing a question of Špalek (2008). Our final contribution is to reprove several Markov-type inequalities from approximation theory by constructing explicit dual solutions to natural linear programs. These inequalities underly the proofs of many of the best-known approximate degree lower bounds, and have important uses throughout theoretical computer science.

Theory of 3-rotation symmetric cubic Boolean functions

Abstract Rotation symmetric Boolean functions have been extensively studied in the last 15 years or so because of their importance in cryptography and coding theory. Until recently, very little was known about such basic questions as when two such functions are affine equivalent. This question in important in applications, because almost all important properties of Boolean functions (such as Hamming weight, nonlinearity, etc.) are affine invariants, so when searching a set for functions with useful properties, it suffices to consider just one function in each equivalence class. This can greatly reduce computation time. Even for quadratic functions, the analysis of affine equivalence was only completed in 2009. The much more complicated case of cubic functions was completed in the special case of affine equivalence under permutations for monomial rotation symmetric functions in two papers from 2011 and 2014. There has also been recent progress for some special cases for functions of degree > 3 ${> 3}$ . In 2007 it was found that functions satisfying a new notion of k-rotation symmetry for k > 1 (where the case k = 1 is ordinary rotation symmetry) were of substantial interest in cryptography and coding theory. Since then several researchers have used these functions for k = 2 and 3 to study such topics as construction of bent functions, nonlinearity and covering radii of various codes. In this paper we develop a detailed theory for the monomial 3-rotation symmetric cubic functions, extending earlier work for the case k = 2 of these functions.

Characterization of robust immune symmetric boolean functions

Fix a field �� $\mathbb {F}$ . The algebraic immunity over �� $\mathbb {F}$ of boolean function f : {0, 1} n � {0, 1} is defined as the minimal degree of a nontrivial (multilinear) polynomial g ( x ) � �� [ x 1 , � , x n ] $g(x) \in \mathbb {F}[x_{1}, \ldots , x_{n}]$ such that f(x) is a constant (either 0 or 1) for all x � {0, 1} n satisfying g(x) = 0. Function f is called k r o b u s t i m m u n e if the algebraic immunity of f is always not less than k no matter how one changes the value of f(x) for k ≤ |x| ≤ n � k. For any field �� $\mathbb {F}$ , any integers n, k � 0, we characterize all k robust immune symmetric boolean functions in n variables. The proof is based on a known symmetrization technique and constructing a partition of nonnegative integers satisfying certain (in)equalities about p-adic distance, where p is the characteristic of the field �� $\mathbb {F}$ .

Families of rotation symmetric functions with useful cryptographic properties

It is known that the set of rotation symmetric Boolean functions has many functions with various useful properties for cryptography. This study shows how to construct some families of rotation symmetric functions which are balanced or plateaued. The authors also consider vectorial Boolean functions [that is, maps from GF (2) n to GF (2) m ] which are k-rotation symmetric and they give two infinite families of such functions which are permutations with the maximum possible algebraic degree. The families of functions that they give provide a source, which can be searched for functions with other useful cryptographic properties.

Permutation equivalence of cubic rotation symmetric Boolean functions

Rotation symmetric Boolean functions have been extensively studied for about 15 years because of their applications in cryptography and coding theory. Until recently little was known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009. For the much more complicated case of cubic rotation symmetric functions generated by a single monomial, the affine equivalence classes under permutations which preserve rotation symmetry were determined in 2011. It was conjectured then that the cubic equivalence classes are the same if all nonsingular affine transformations, not just permutations, are allowed. This conjecture is probably difficult, but here we take a step towards it by proving that the cubic affine equivalence classes found in 2011 are the same if all permutations, not just those preserving rotation symmetry, are allowed. The needed new idea uses the theory of circulant matrices.

Construction of RSBFs with improved cryptographic properties to resist differential fault attack on grain family of stream ciphers

In recent literature, the differential fault analysis (DFA) on Grain family of stream ciphers has been shown to exploit the low algebraic degree of the derivative of the nonlinear combining function h of the stream cipher, h(x) ? h(x ? ?). The low algebraic degree allows the DFA adversary to create a linearly independent system of equations generated from the faulty and fault-free keystreams and use these equations to extract the initial state of the NFSR and LFSR stages in the stream cipher. In this paper, we propose a construction scheme for rotation symmetric Boolean functions (RSBFs) h(x) along with an orbit-tuple flip based iterative hill-climbing based construction algorithm for balanced RSBFs with high nonlinearity, low absolute indicator value of global avalanche characteristics (GAC), and high algebraic degree of h(x) ? h(x ? ?). The construction algorithm is scalable for higher input variables like n = 9,10,11 as shown in the paper. We find some interesting autocorrelation spectra and Walsh spectra properties for the class of RSBFs and then use them in the construction of RSBFs with improved cryptographic properties. We present the cryptographic properties of the RSBFs constructed for high input variables which can be used to make DFA attack harder using the existing techniques.

Constructions of resilient rotation symmetric Boolean functions on given number of variables

In this study, the properties of the support tables of rotation symmetric Boolean functions (RSBFs for simplicity) are studied, and two sufficient and necessary conditions for RSBFs being 1- and 2-resilient are obtained, respectively. Based on the relations between resilient functions and orthogonal arrays, with the help of the properties about the support tables of RSBFs, it is shown that the constructions of 1-resilient RSBFs on given number of variables are equivalent to solving an equation system, and the number of functions is equal to the number of solutions of the equation system. Moreover, similar results are also obtained for 2-resilient RSBFs. Lastly, a simple example is given to demonstrate our method. The results indicate that the constructions of n-variable 1-resilient RSBFs are equivalent to studying the cyclotomic cosets Cs modulo 2 n − 1 with respect to 2.

Construction of balanced rotation symmetric Boolean functions with optimal algebraic immunity

Algebraic immunity is a new cryptographic criterion proposed against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in many stream ciphers should possess high algebraic immunity. This paper presents one main result to find balanced rotation symmetric Boolean functions with maximum algebraic immunity. Through swapping the values of two orbits of rotation class of the majority function, a class of 4k+1 variable Boolean functions with maximum algebraic immunity is constructed. The function f(x) we construct always has terms of degree n-2 independence of what ever n is. And the nonlinearity of f(x) is relatively good for large n.

Efficient quantum algorithms to construct arbitrary Dicke states

In this paper, we study several quantum algorithms toward the efficient construction of arbitrary arbitrary Dicke state. The proposed algorithms use proper symmetric Boolean functions that involve manipulation with Krawtchouk polynomials. Deutsch---Jozsa algorithm, Grover algorithm, and the parity measurement technique are stitched together to devise the complete algorithm. In addition to that we explore how the biased Hadamard transformation can be utilized into our strategy, motivated by the work of Childs et al. (Quantum Inf Comput 2(3):181---191, 2002).

Asymptotic Behavior of Perturbations of Symmetric Functions

In this paper we consider perturbations of symmetric Boolean functions \({{\sigma_{n,k_1}} +\ldots+{\sigma_{n,k_s}}}\) in n-variable and degree ks. We compute the asymptotic behavior of Boolean functions of the type $${\sigma_{n,k_1}} +\ldots+{\sigma_{n,k_s}} +F(X_1, . . . , X_j)$$ for j fixed. In particular, we characterize all the Boolean functions of the type $${\sigma_{n,k_1}} +\ldots+{\sigma_{n,k_s}} +F(X_1, . . . , X_j)$$ that are asymptotic balanced. We also present an algorithm that computes the asymptotic behavior of a family of Boolean functions from one member of the family. Finally, as a byproduct of our results, we provide a relation between the parity of families of sums of binomial coefficients.

Algebraic Immunity, Correlation Immunity and other Cryptographic Properties of Quadratic Rotation Symmetric Boolean Functions

Boolean functions with a variety of secure cipher properties are the key factors to design cryptosystem with the ability to resist multiple cipher attacks and good safety performance. In this paper, using the derivative of the Boolean functions and the e-derivative defined by ourselves as the main research tools, we study algebraic immunity, correlation immunity and other cryptographic properties of the quadratic rotation symmetric Boolean functions. We determine the quadratic rotation symmetric Boolean functions which are H Boolean functions, and the range of weight distribution of the quadratic rotation symmetry H Boolean functions. Besides, we get the compatibility among propagation, balance, correlation immunity and algebraic immunity of the quadratic rotation symmetry H Boolean functions, and also focus on the relationship of balance, correlation immunity and dimension. Furthermore, we check the existence of the cubic rotation symmetry H Boolean functions, and obtain the relationship between existence and dimension of the cubic rotation symmetry H Boolean functions. Moreover, we obtain a more convenient method for solving annihilator. Such researches are important in cryptographic primitive designs, and have significance and role in the theory and application range of cryptosystems.

The conjunction complexity asymptotic of self-correcting circuits for monotone symmetric functions with threshold 2

It is shown that the conjunction complexity Lk&(f2n) of monotone symmetric Boolean functions \(f_2^n (x_1 , \ldots ,x_n ) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n} x_i x_j\) realized by k-self-correcting circuits in the basis B = {&, −} asymptotically equals to (k + 2)n for growing n providing the price of a reliable conjunctor is ≥ k + 2.

Upper bounds for the formula size of symmetric Boolean functions

We prove that the complexity of the implementation of the counting function of n Boolean variables by binary formulas is at most n3.03, and it is at most n4.47 for DeMorgan formulas. Hence, the same bounds are valid for the formula size of any threshold symmetric function of n variables, particularly, for the majority function. The following bounds are proved for the formula size of any symmetric Boolean function of n variables: n3.04 for binary formulas and n4.48 for DeMorgan ones. The proof is based on the modular arithmetic.

Affine equivalence for cubic rotation symmetric Boolean functions with [formula omitted] variables

Rotation symmetric Boolean functions have been extensively studied in the last fifteen years or so because of their importance in cryptography and coding theory. Until recently, very little was known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in a 2009 paper of Kim, Park and Hahn. The much more complicated analogous problem for cubic functions was solved for permutations using a new concept of patterns in a 2011 paper of Cusick, and it is conjectured that, as in the quadratic case, this solution actually applies for all affine transformations. The patterns method enables a detailed analysis of the affine equivalence classes for various special classes of cubic rotation symmetric functions in n variables. Here the case of functions generated by a single monomial and having pq variables, where p and q are primes, is examined in detail, and in particular, a formula for the number of classes is proved. This is significant because it is the first time that a complete enumeration of the number of classes has been found when the number of variables is divisible by two distinct primes.

Distinguishing properties and applications of higher order derivatives of Boolean functions

Higher order differential cryptanalysis is based on a property of higher order derivatives of Boolean functions such that derivative of a Boolean function reduces its degree at least 1 and continuously taking derivatives eventually yields a zero function. A quicker degree reduction means a lower data complexity in cryptanalysis, which can be determined by fast point at which the derivative reduces the degree at least 2.In this paper, we show that the set of the fast points of a Boolean function constitutes a linear subspace and its dimension plus the degree of the function is at most the size of the function. We also show that non-zero fast point exists in every n-variable Boolean function of degree n-1, every symmetric Boolean function of degree d where n≢d(mod2) or every quadratic Boolean function of odd number variables, which help us distinguish a few block ciphers and propose a new design principle about degree for block cipher.

On the minimal fourier degree of symmetric Boolean functions

In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function. Specifically, we prove that for every non-linear and sym...

New Construction of Even-variable Rotation Symmetric Boolean Functions with Optimum Algebraic Immunity

The rotation symmetric Boolean functions which are invariant under the action of cyclic group have been used as components of different cryptosystems. In order to resist algebraic attacks, Boolean functions should have high algebraic immunity. This paper studies the construction of even-variable rotation symmetric Boolean functions with optimum algebraic immunity. We construct

Counting rotation symmetric functions using Polya’s theorem

Homogeneous rotation symmetric (invariant under cyclic permutation of the variables) Boolean functions have been extensively studied in recent years due to their applications in cryptography. In this paper we give an explicit formula for the number of homogeneous rotation symmetric functions over the finite field GF(pm) using Polya’s enumeration theorem, which completely solves the open problem proposed by Yuan Li in 2008. This result simplifies the proof and the nonexplicit counting formula given by Shaojing Fu et al. over the field GF(p). This paper also gives an explicit count for n-variable balanced rotation symmetric Boolean functions with n=pq, where p and q are distinct primes. Previous work only gave an explicit count for the case where n is prime and lower bounds for the case where n is a prime power.

The Degree of Two Classes of 3rd Order Correlation Immune Symmetric Boolean Functions

The Degree of Two Classes of 3rd Order Correlation Immune Symmetric Boolean Functions