Modified Patterson–Wiedemann construction

Abstract

The Patterson–Wiedemann (PW) construction, which is defined for an odd number n of variables with n being the product of two distinct prime numbers p and q, can be interpreted as idempotent functions which are represented by the (d, r)-interleaved sequences formed by all-zero and all-one columns, where $r=(2^p-1)(2^q-1)$ and $d=\frac{(2^n-1)}{r}$. We here study a modified form of the PW construction, which only requires $2^n-1$ $(= dr)$ be a composite number, by relaxing the constraint on the values of d and r. We first elaborate on the case $n=15$ and consider the functions corresponding to the (217, 151)-interleaved sequences. Taking into account those satisfying $f(\alpha ) = f(\alpha ^{2^k})$ for all $\alpha \in \mathbb {F}_{2^{n}}$ in this scenario, where k is a fixed divisor of n, we obtain Boolean functions with nonlinearity 16268 exceeding the bent concatenation bound. Then we extend our study for the case $n=11$ and obtain Boolean functions with nonlinearity 996 represented by the (89, 23)-interleaved sequences, which equals the best known nonlinearity result. In the process, we show that there is the possibility to exceed the best known nonlinearities using the functions corresponding to those interleaved sequences.