On third-order nonlinearity of biquadratic monomial Boolean functions (Q275994)

Summary: The \(r\)th-order nonlinearity of Boolean function plays a central role against several known attacks on stream and block ciphers. Because of the fact that its maximum equals the covering radius of the \(r\)th-order Reed-Muller code, it also plays an important role in coding theory. The computation of exact value or high lower bound on the \(r\)th-order nonlinearity of a Boolean function is very complicated problem, especially when \(r>1\). This paper is concerned with the computation of the lower bounds for third-order nonlinearities of two classes of Boolean functions of the form \(\mathrm{Tr}^n_1(\lambda x^d)\) for all \(x \in \mathbb F_{2^n}\), \(\lambda \in \mathbb F^*_{2^n}\), where (a) \(d=2^i+2^j+2^k+1\), where \(i\), \(j\), and \(k\) are integers such that \(i>j>k\geq 1\) and \(n>2i\), and (b) \(d=2^{3\ell}+2^{2\ell}+2^{\ell}+1\), where \(\ell\) is a positive integer such that \(\mathrm{gcd}(\ell,n)=1\) and \(n>6\).