Patterson-Wiedemann type functions on 21 variables with Nonlinearity greater than Bent Concatenation bound

Nonlinearity is one of the most challenging combinatorial property in the domain of Boolean function research. Obtaining nonlinearity greater than the bent concatenation bound for odd number of variables continues to be one of the most sought after combinatorial research problems. The pioneering result in this direction has been discovered by Patterson and Wiedemann in 1983 (IEEE-IT), which considered Boolean functions on $5 \times 3 = 15$ variables that are invariant under the actions of the cyclic group ${GF(2^5)}^\ast \cdot {GF(2^3)}^\ast$ as well as the group of Frobenius authomorphisms. Some of these Boolean functions posses nonlinearity greater than the bent concatenation bound. The next possible option for exploring such functions is on $7 \times 3 = 21$ variables. However, obtaining such functions remained elusive for more than three decades even after substantial efforts as evident in the literature. In this paper, we exploit combinatorial arguments together with heuristic search to demonstrate such functions for the first time

Balanced Boolean Functions with (Almost) Optimal Algebraic Immunity and Very High Nonlinearity

In this paper, we present a class of $2k$-variable balanced Boolean functions and a class of $2k$-variable $1$-resilient Boolean functions for an integer $k\ge 2$, which both have the maximal algebraic degree and very high nonlinearity. Based on a newly proposed conjecture by Tu and Deng, it is shown that the proposed balanced Boolean functions have optimal algebraic immunity and the $1$-resilient Boolean functions have almost optimal algebraic immunity. Among all the known results of balanced Boolean functions and $1$-resilient Boolean functions, our new functions possess the highest nonlinearity. Based on the fact that the conjecture has been verified for all $k\le 29$ by computer, at least we have constructed a class of balanced Boolean functions and a class of $1$-resilient Boolean functions with the even number of variables $\le 58$, which are cryptographically optimal or almost optimal in terms of balancedness, algebraic degree, nonlinearity, and algebraic immunity

Idempotents in the Neighbourhood of Patterson-Wiedemann Functions having Walsh Spectra Zeros

In this paper we study the neighbourhood of 15-variable Patterson-Wiedemann (PW) functions, i.e., the functions that differ by a small Hamming distance from the PW functions in terms of truth table representation. We exploit the idempotent structure of the PW functions and interpret them as Rotation Symmetric Boolean Functions (RSBFs). We present techniques to modify these RSBFs to introduce zeros in the Walsh spectra of the modified functions with minimum reduction in nonlinearity. Our technique demonstrates 15-variable balanced and 1-resilient functions with currently best known nonlinearities 16272 and 16264 respectively. In the process, we find functions for which the autocorrelation spectra and algebraic immunity parameters are best known till date