The focus of this article is on nonlinear characteristics of cryptographic Boolean functions. First, we introduce the notion of plateaued functions that have many cryptographically desirable properties. Second, we establish a sequence of strengthened inequalities on some of the most important nonlinearity criteria, including nonlinearity, avalanche, and correlation immunity, and prove that critical cases of the inequalities coincide with characterizations of plateaued functions. We then proceed to prove that plateaued functions include as a proper subset all partially bent functions that were introduced earlier by Claude Carlet (1993). This solves an interesting problem that arises naturally from previously known results on partially bent functions. In addition, we construct plateaued, but not partially bent, functions that have many properties useful in cryptography.

This correspondence studies resilient functions which have applications in fault-tolerant distributed computing, quantum cryptographic key distribution, and random sequence generation for stream ciphers. We present a number of new methods for synthesizing resilient functions. An interesting aspect of these methods is that they are applicable both to linear and nonlinear resilient functions. Our second major contribution is to show that every linear resilient function can be transformed into a large number of nonlinear resilient functions with the same parameters. As a result, we obtain resilient functions that are highly nonlinear and have a high algebraic degree.

We prove that the conjecture on Bent sequences stated in the paper written by Kumar, Scholtz and Welch (see J. Combinatorial Theory, Ser A, vol.40, p.90-107, 1985) is false.>

Substitution boxes (S-boxes) are a crucial component of DES-like block ciphers. This research addresses problems with previous approaches towards constructing S-boxes, and proposes a new definition for the robustness of S-boxes to differential cryptanalysis, which is the most powerful cryptanalytic attack known to date. A novel method based on group Hadamard matrices is developed to systematically generate S-boxes that satisfy a number of critical cryptographic properties. Among the properties are the high nonlinearity, the strict avalanche characteristics, the balancedness, the robustness against differential cryptanalysis, and the immunity to linear cryptanalysis. An example is provided to illustrate the S-box generating method.