Modifying Bent Functions to Obtain the Balanced Ones with High Nonlinearity

Abstract

Balanced Boolean functions with high nonlinearity are considered as major cryptographic primitives in the design of symmetric key cryptosystems. Dobbertin, in early nineties, gave an explicit construction for balanced functions on (even) n variables, with nonlinearity $2^{n-1}-2^{\frac{n}{2}} + nlb(\frac{n}{2})$, where nlb(t) is the maximum nonlinearity of a balanced Boolean functions in t variables and conjectured that $nlb(n) \le 2^{n-1} - 2^{\frac{n}{2}} + nlb(\frac{n}{2})$. This bound still holds. In this paper we revisit the problem. First we present a detailed combinatorial analysis related to highly nonlinear balanced functions exploiting the inter-related properties like weight, nonlinearity, and Walsh–Hadamard spectrum. Our results provide a general framework to cover the works of Sarkar-Maitra (Crypto 2000), Maity-Johansson (Indocrypt 2002), and Maity-Maitra (FSE 2004) as special cases. In this regard, we revisit the well-known construction methods through modification of bent functions and provide supporting examples for 8, 10, 12, and 14 variables. We believe these results will advance the understanding related to highly nonlinear balanced Boolean functions on even numbers of variables as well as the Dobbertin’s conjecture.