Maiorana-McFarland class: Degree optimization and algebraic properties

Enes Pasalic

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    Abstract

    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) -> 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.
    Original languageEnglish
    JournalI E E E Transactions on Information Theory
    Volume52
    Issue number10
    Pages (from-to)4581-4594
    ISSN0018-9448
    DOIs
    Publication statusPublished - 2006

    Bibliographical note

    Copyright: 2006 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE

    Keywords

    • Boolean function
    • algebraic degree
    • nonlinearity
    • resiliency
    • algebraic immunity
    • vectorial Boolean function

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