New Construction for Balanced Boolean Functions with Very High Nonlinearity

In the past twenty years, there were only a few constructions for Boolean functions with nonlinearity exceeding the quadratic bound 2n−1 − 2(n−1)/2 when n is odd (we shall call them Boolean functions with very high nonlinearity). The first basic construction was by Patterson and Wiedemann in 1983, which produced unbalanced function with very high nonlinearity. But for cryptographic applications, we need balanced Boolean functions. Therefore in 1993, Seberry, Zhang and Zheng proposed a secondary construction for balanced functions with very high nonlinearity by taking the direct sum of a modified bent function with the Patterson-Wiedemann function. Later in 2000, Sarkar and Maitra constructed such functions by taking the direct sum of a bent function with a modified Patterson-Wiedemann function. In this paper, we propose a new secondary construction for balanced Boolean functions with very high nonlinearity by recursively composing balanced functions with very high nonlinearity with quadratic functions. This is the first construction for balanced function with very high nonlinearity not based on the direct sum approach. Our construction also have other desirable properties like high algebraic degree and large linear span. key words: balanced Boolean functions, high nonlinearity, cascaded GMW sequences, Patterson-Wiedemann construction