Maiorana-McFarland Class: Degree Optimization and Algebraic Properties

In this paper, we consider a subclass of the Maiorana–McFarland class used in the design of resilient nonlinear Boolean functions. We show that these functions allow a simple modification so that resilient Boolean functions of maximum algebraic degree may be generated instead of suboptimized degree in the original class. Preserving a high-nonlinearity value immanent to the original construction method, together with the degree optimization gives in many cases functions with cryptographic properties superior to all previously known construction methods. This approach is then used to increase the algebraic degree of functions in the extended Maiorana–McFarland (MM) class (nonlinear resilient functions F : GF (2)n 7! GF(2)m derived from linear codes). We also show that in the Boolean case, the same subclass seems not to have an optimized algebraic immunity, hence not providing a maximum resistance against algebraic attacks. A theoretical analysis of the algebraic properties of extended Maiorana–McFarland class indicates that this class of functions should be avoided as a filtering function in nonlinear combining generators.