Symmetric Boolean Functions Research Articles

Constructions of 2-Correlation Immune Rotation Symmetric Boolean Functions

Constructions of 2-Correlation Immune Rotation Symmetric Boolean Functions

Further construction of even-variable balanced rotation symmetric Boolean functions with optimal algebraic immunity

The field of cryptography has given a lot of attention to rotation symmetric Boolean functions (RSBFs) because they possess special structures and include many functions with good cryptographic properties. It is difficult to construct even-variable balanced RSBFs with optimal algebraic immunity in the study on RSBFs. Recently, Mesnager et al. proposed the first and only construction of balanced RSBFs with optimal algebraic immunity for arbitrary even variables (Des. Codes Cryptogr. 89 (1) (2021) 1-17). The nonlinearity of their functions is not high. In this paper, we develop further research based on their construction and present a fresh design of n-variable balanced RSBFs with optimal algebraic immunity for arbitrary even n. Our functions have higher nonlinearity compared to their functions. Furthermore, the algebraic degree and fast algebraic immunity of our functions are not less than n−2 for n not bigger than 16.

Constructions of 2-resilient rotation symmetric Boolean functions with odd number of variables

In this paper, a new method for constructing 2-resilient rotation symmetric Boolean functions with odd number of variables is presented. Based on an equivalent characterization of this class of functions and the relation between the orbit matrices and their complements, a system of equations about 2-tuples distribution matrix is established. Then the constructions of some 2-resilient rotation symmetric Boolean functions with odd number of variables can be converted into the solutions of the system of equations. We also give a sufficient condition for the constructed functions to have the maximum algebraic degree. Moreover, we provide a lower bound on the number of nonlinear 2-resilient rotation symmetric Boolean functions with odd number of variables. Particularly, the lower bound is 152 for seven number of variables.

Short [formula omitted]-rotation symmetric Boolean functions

Rotation symmetric Boolean functions were introduced by Pieprzyk and Qu in the late 1990’s. They showed that these functions are useful, among other things, in the design of fast hashing algorithms with strong cryptographic properties. The concept of rotation symmetric Boolean functions has been generalized to a class of functions known as k-rotation Boolean functions, where k divides n and n is the number of variables of the Boolean function. Analogous to the case of regular rotation symmetric Boolean functions, a monomial k-rotation Boolean function is called long k-cycle if the number of terms coincides with n/k and short k-cycle if the number of terms is less than n/k. In this work we characterize short k-cycles by providing specific generators for them. We also provide a count for the number of short k-cycles in n variables for a fixed degree and for a specified length.

A New Characterization of 2-Resilient Rotation Symmetric Boolean Functions

A New Characterization of 2-Resilient Rotation Symmetric Boolean Functions

Balanced Even-Variable Rotation Symmetric Boolean Functions with Optimal Algebraic Immunity, Maximum Algebraic Degree and Higher Nonlinearity

Rotation symmetric Boolean functions are good candidates for stream ciphers because they have such advantages as simple structure, high operational speed and low implement cost. Recently, Mesnager et al. proposed for the first time an efficient method to construct balanced rotation symmetric Boolean functions with optimal algebraic immunity and good nonlinearity for an arbitrary even number of variables. However, the algebraic degree of their constructed [Formula: see text]-variable ([Formula: see text]) function is always less than the maximum value [Formula: see text]. In this paper, by modifying the support of Boolean functions from Mesnager et al.’s construction, we present two new constructions of balanced even-variable rotation symmetric Boolean functions with optimal algebraic immunity as well as higher algebraic degree and nonlinearity. The algebraic degree of Boolean functions in the first construction reaches the maximum value [Formula: see text] if [Formula: see text] is odd and [Formula: see text] or [Formula: see text] for integer [Formula: see text], while that of the second construction reaches the maximum value for all [Formula: see text]. Moreover, the nonlinearities of Boolean functions in both two constructions are higher than that of Mesnager et al.’s construction.

Balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity and higher nonlinearity

By modifying the support of the majority function, we propose a construction of balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity. For n ≥ 17 , the nonlinearity of these constructed functions is the highest among all known rotation symmetric Boolean functions with optimal algebraic immunity. The algebraic degree can reach the maximum for some special numbers of variables. Moreover, simulations show that such functions have good fast algebraic immunity for some small numbers of variables.

Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms

We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in mathsf {Quasi}text {-}mathsf {NP} = mathsf {NTIME}[n^{(log n)^{O(1)}}] and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes mathcal { C}, by showing that mathcal { C} admits non-trivial satisfiability and/or # SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial # SAT algorithm for a circuit class {mathcal C}. Say that a symmetric Boolean function f(x1,…,xn) is sparse if it outputs 1 on O(1) values of {sum }_{i} x_{i}. We show that for every sparse f, and for all “typical” mathcal { C}, faster # SAT algorithms for mathcal { C} circuits imply lower bounds against the circuit class f circ mathcal { C}, which may be stronger than mathcal { C} itself. In particular: # SAT algorithms for nk-size mathcal { C}-circuits running in 2n/nk time (for all k) imply NEXP does not have (f circ mathcal { C})-circuits of polynomial size.# SAT algorithms for 2^{n^{{varepsilon }}}-size mathcal { C}-circuits running in 2^{n-n^{{varepsilon }}} time (for some ε > 0) imply Quasi-NP does not have (f circ mathcal { C})-circuits of polynomial size.Applying # SAT algorithms from the literature, one immediate corollary of our results is that Quasi-NP does not have EMAJ ∘ ACC0 ∘ THR circuits of polynomial size, where EMAJ is the “exact majority” function, improving previous lower bounds against ACC0 [Williams JACM’14] and ACC0 ∘THR [Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.

Certificate complexity of elementary symmetric Boolean functions

Boolean functions have important applications in information technology and computer science. Certificate complexity is an important combinatorial measure of Boolean function complexity. In this paper, we first introduce many concise and efficient notations then we study the elementary symmetric Boolean functions and obtain their certificate complexities when their degrees are odd or powers of 2. We show that both upper and lower bounds of certificate complexities can be attained.

Recursions for modified Walsh transforms of some families of Boolean functions

We show that, under certain conditions, restricted and biased exponential sums and Walsh transforms of symmetric and rotation symmetric Boolean functions are, as in the case of nonbiased domain, C-finite sequences. We also prove that under other conditions, these sequences are P-finite, which is a somewhat different behavior than their nonbiased counterparts. We further show that exponential sums and Walsh transforms of a family of rotation symmetric monomials over the restricted domain En,j={x∈𝔽2n:wt (x)=j} (wt (x) is the weight of the vector x) are given by polynomials of degree at most j, and so, they are also C-finite sequences. Finally, we also present a study of the behavior of symmetric Boolean functions under these biased transforms.

The linear structures and fast points of rotation symmetric Boolean functions

The linear structures and fast points of rotation symmetric Boolean functions

Symbolic dynamics and rotation symmetric Boolean functions

We identify the weights wt(fn) of a family {fn} of rotation symmetric Boolean functions with the cardinalities of the sets of n-periodic points of a finite-type shift, recovering the second author’s result that said weights satisfy a linear recurrence. Similarly, the weights of idempotent functions fn defined on finite fields can be recovered as the cardinalities of curves over those fields and hence satisfy a linear recurrence as a consequence of the rationality of curves’ zeta functions. Weil’s Riemann hypothesis for curves then provides additional information about wt(fn). We apply our results to the case of quadratic functions and considerably extend the results in an earlier paper of ours.

Constructions of 2-resilient rotation symmetric Boolean functions through symbol transformations of cyclic Hadamard matrix

In this paper, the properties of symbol transformations of cyclic Hadamard matrices are studied. An infinite class of (n−1)-variable 2-resilient rotation symmetric Boolean functions are constructed, and the nonlinearity of the constructed functions is 2n(n−1). The crucial technique of this method is to determine a subset T⊆F2n−1 satisfying a correspondent condition. This is a new construction of 2-resilient rotation symmetric Boolean functions via switching the supports of (n−1)-variable rotation symmetric Boolean functions of degree one, i.e., f0n−1(x1,x2,⋯,xn−1)=⊕i=1n−1xi, where n=4t.

Searching for Cryptographically Significant Rotation Symmetric Boolean Functions by Designing Heuristic Algorithms

It has been proved that the set of rotation symmetric Boolean functions (RSBFs) is abundant in cryptographically strong functions with multiple criteria. In this study, we design two genetic algorithms and apply them to search for balanced RSBFs with high nonlinearity. The experimental results show that our methods can generate cryptographically strong Boolean functions with high nonlinearity, 1-resilient functions, and optimal algebraic immunity. It shows that these functions have superiority from the view point of practical application in cryptosystems compared with known ones which are obtained by other heuristics.

A Novel Construction of 2-Resilient Rotation Symmetric Boolean Functions

A Novel Construction of 2-Resilient Rotation Symmetric Boolean Functions

The Stochastic Boolean Function Evaluation problem for symmetric Boolean functions

We give two approximation algorithms solving the Stochastic Boolean Function Evaluation (SBFE) problem for symmetric Boolean functions. The first is an O(logn)-approximation algorithm, based on the submodular goal-value approach of Deshpande, Hellerstein and Kletenik. Our second algorithm, which is simple, is based on the algorithm solving the SBFE problem for k-of-n functions, due to Salloum, Breuer, and Ben-Dov. It achieves a (B−1) approximation factor, where B is the number of blocks of 0’s and 1’s in the standard vector representation of the symmetric Boolean function. As part of the design of the first algorithm, we prove that the goal value of any symmetric Boolean function is less than n(n+1)/2. Finally, we give an example showing that for symmetric Boolean functions, minimum expected verification cost and minimum expected evaluation cost are not necessarily equal. This contrasts with a previous result, given by Das, Jafarpour, Orlitsky, Pan and Suresh, which showed that equality holds in the unit-cost case.

Several classes of even-variable 1-resilient rotation symmetric Boolean functions with high algebraic degree and nonlinearity

Recent research shows that the class of rotation symmetric Boolean functions is potentially rich in functions of cryptographic significance. In this paper, some classes of 2m-variable (m is an odd integer) 1-resilient rotation symmetric Boolean functions are got, whose nonlinearity and algebraic degree are studied. For the first time, we obtain 2m-variable 1-resilient rotation symmetric Boolean functions having high nonlinearity and optimal algebraic degree. In addition, we obtain a class of non-linear rotation symmetric 1-resilient function for every n≥5, and a class of quadratic rotation symmetric (k−1)-resilient function of n=3k variables, where k is an integer.

A Method of Realizing XOR/XNOR Gate using Symmetric Boolean Function Lattice Structure

The current CMOS’s industry standard XOR and XNOR gate consist of 12 and 10 transistors, respectively. This transistor count could be lowered down to produce low power circuits as XOR/XNOR are extensively used in many functional modules. As a solution, a method for realizing low transistor count XOR/XNOR gates using a special property of symmetric Boolean function is proposed. This property suggests that the circuits for such functions can be realized with fewer transistors using a special lattice structure circuit. Modifications are made to the original lattice structure to match with the current CMOS technology requirements. The final circuits require eight transistors each for XOR/XNOR with mixtures of NMOS and PMOS at push-up and pull-down networks. Simulations show that the intended logic functions of XOR/XNOR are achieved. The reading of actual voltage swing, however, shows that the output is either 0.3 V over ground or below VDD when there is a mixture of NMOS and PMOS as pull-down or push-up networks, respectively. More voltage loss of 0.4 V is observed if only NMOS is at push-up or only PMOS is at pull-down networks. As a preliminary work, this achievement of the functional logic level warrants more future work to improve the loss in output voltage swing.

On the complexity of monotone circuits for threshold symmetric Boolean functions

Abstract The complexity of implementation of a threshold symmetric n-place Boolean function with threshold k = O(1) via circuits over the basis {∨, ∧} is shown not to exceed 2 log2 k ⋅ n + o(n). Moreover, the complexity of a threshold-2 function is proved to be 2n + Θ( $\begin{array}{} \sqrt n \end{array} $ ), and the complexity of a threshold-3 function is shown to be 3n + O(log n), the corresponding lower bounds are put forward.

Implementation of the method of figurative transformations to minimizing symmetric Boolean functions

This paper reports a study that has established the possibility of improving the effectiveness of the method of figurative transformations in order to minimize symmetrical Boolean functions in the main and polynomial bases. Prospective reserves in the analytical method were identified, such as simplification of polynomial function conjuncterms using the created equivalent transformations based on the method of inserting the same conjuncterms followed by the operation of super-gluing the variables. The method of figurative transformations was extended to the process of minimizing the symmetrical Boolean functions with the help of algebra in terms of rules for simplifying the functions of the main and polynomial bases and developed equivalent transformations of conjuncterms. It was established that the simplification of symmetric Boolean functions by the method of figurative transformations is based on a flowchart with repetition, which is the actual truth table of the assigned function. This is a sufficient resource to minimize symmetrical Boolean functions that makes it possible to do without auxiliary objects, such as Karnaugh maps, cubes, etc. The perfect normal form of symmetrical functions can be represented by binary matrices that would represent the terms of symmetrical Boolean functions and the OR or XOR operation for them. The experimental study has confirmed that the method of figurative transformations that employs the 2-(n, b)-design, and 2-(n, x/b)-design combinatorial systems improves the efficiency of minimizing symmetrical Boolean functions. Compared to analogs, this makes it possible to enhance the productivity of minimizing symmetrical Boolean functions by 100‒200 %. There are grounds to assert the possibility of improving the effectiveness of minimizing symmetrical Boolean functions in the main and polynomial bases by the method of figurative transformations. This is ensured, in particular, by using the developed equivalent transformations of polynomial function conjuncterms based on the method of inserting similar conjuncterms followed by the operation of super-gluing the variables.