Related papers: Yet another algorithm to compute the nonlinearity …
We compute the nonlinearity of Boolean functions with Groebner basis techniques, providing two algorithms: one over the binary field and the other over the rationals. We also estimate their complexity. Then we show how to improve our…
An algorithm for computing the nonlinearity of a Boolean function from its algebraic normal form (ANF) is proposed. By generalizing the expression of the weight of a Boolean function in terms of its ANF coefficients, a formulation of the…
The nonlinearity of a Boolean function is a key property in deciding its suitability for cryptographic purposes, e.g. as a combining function in stream ciphers, and so the nonlinearity computation is an important problem for applications.…
We study the computational complexity of two Boolean nonlinearity measures: the nonlinearity and the multiplicative complexity. We show that if one-way functions exist, no algorithm can compute the multiplicative complexity in time…
Non-linearity of a Boolean function indicates how far it is from any linear function. Despite there being several strong results about identifying a linear function and distinguishing one from a sufficiently non-linear function, we found a…
Classification of Non-linear Boolean functions is a long-standing problem in the area of theoretical computer science. In this paper, effort has been made to achieve a systematic classification of all n-variable Boolean functions, where…
We present new algorithms to compute fundamental properties of a Boolean function given in truth-table form. Specifically, we give an O(N^2.322 log N) algorithm for block sensitivity, an O(N^1.585 log N) algorithm for `tree decomposition,'…
Bernstein-Vazirani algorithm (the one-query algorithm) can identify a completely specified linear Boolean function using a single query to the oracle with certainty. The first aim of the paper is to show that if the provided Boolean…
In this paper we study the complexity of quantum query algorithms computing the value of Boolean function and its relation to the degree of algebraic polynomial representing this function. We pay special attention to Boolean functions with…
The r-th order nonlinearity of a Boolean function is the minimum number of elements that have to be changed in its truth table to arrive at a Boolean function of degree at most r. It is shown that the (suitably normalised) r-th order…
We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main…
We consider the multiplicative complexity of Boolean functions with multiple bits of output, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For so-called $\Sigma\Pi\Sigma$…
We present a new algorithm by which the Adomian polynomials can be determined for scalar-valued nonlinear polynomial functional in a Hilbert space. This algorithm calculates the Adomian polynomials without the complicated operations such as…
The algebraic degree of Boolean functions (or vectorial Boolean functions) is an important cryptographic parameter that should be computed by fast algorithms. They work in two main ways: (1) by computing the algebraic normal form and then…
This paper investigates the learnability of the nonlinearity property of Boolean functions using neural networks. We train encoder style deep neural networks to learn to predict the nonlinearity of Boolean functions from examples of…
We propose a quantum algorithm to estimate the Gowers $U_2$ norm of a Boolean function, and extend it into a second algorithm to distinguish between linear Boolean functions and Boolean functions that are $\epsilon$-far from the set of…
Boolean functions are mathematical objects used in diverse applications. Different applications also have different requirements, making the research on Boolean functions very active. In the last 30 years, evolutionary algorithms have been…
The parity decision tree model extends the decision tree model by allowing the computation of a parity function in one step. We prove that the deterministic parity decision tree complexity of any Boolean function is polynomially related to…
Here we consider an approach for fast computing the algebraic degree of Boolean functions. It combines fast computing the ANF (known as ANF transform) and thereafter the algebraic degree by using the weight-lexicographic order (WLO) of the…
We discuss quantum algorithms, based on the Bernstein-Vazirani algorithm, for finding which variables a Boolean function depends on. There are 2^n possible linear Boolean functions of n variables; given a linear Boolean function, the…