Degree Optimized Resilient Boolean Functions from Maiorana-McFarland Class

Abstract

AbstractIn this paper we present a construction method of degree optimized resilient Boolean functions with very high nonlinearity. We present a general construction method valid for any n ≥ 4 and for order of resiliency t satisfying t ≤ n-3. The construction is based on the modification of the famous Marioana-McFarland class in a controlled manner such that the resulting functions will contain some extra terms of high algebraic degree in its ANF including one term of highest algebraic degree. Hence, the linear complexity is increased, the functions obtained reach the Siegentheler’s bound and furthermore the nonlinearity of such a function in many cases is superior to all previously known construction methods. This construction method is then generalized to the case of vectorial resilient functions, that is {F} : \(\mathbb F{^n_2}\)↦\(\mathbb F{^m_2}\), providing functions of very high algebraic degree almost reaching the Siegenthaler’s upper bound.KeywordsBoolean FunctionResiliencyNonlinearityAlgebraic Degree