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On Bent and Highly Nonlinear Balanced/Resilient Functions and Their Algebraic Immunities

  • Conference paper
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3857))

Abstract

Since the introduction of the notions of nonlinearity in the mid-70’s (the term has been in fact introduced later), of correlation immunity and resiliency in the mid-80’s, and of algebraic immunity recently, the problem of efficiently constructing Boolean functions satisfying, at high levels, one or several of these criteria has received much attention. Only few primary constructions are known, and secondary constructions are also necessary to obtain functions achieving or approaching the best possible cryptographic characteristics. After recalling the background on cryptographic criteria and making some general observations, we try to give a survey of all these constructions and their properties. We then show that a nice and simple property of Boolean functions leads to a general secondary construction building an n-variable function from three known n-variable functions. This construction generalizes secondary constructions recently obtained for Boolean bent functions and also leads to secondary constructions of highly nonlinear balanced or resilient functions, with potentially better algebraic immunities than the three functions used as building blocks.

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Authors and Affiliations

  1. Projet CODES, INRIA, BP 105, 78153 Cedex, Le Chesnay, France

    Claude Carlet

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  1. Claude Carlet
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, University of Hawaii, Honolulu, USA

    Marc P. C. Fossorier

  2. National Institute of Advanced Industrial Science and Technology (AIST), Research Center for Information Security (RCIS),  

    Hideki Imai

  3. Department of Electrical and Computer Engineering, University of California, CA 95616, Davis,  

    Shu Lin

  4. University Paul Sabatier, AAECC/IRIT, 118 route de Narbonne, 31300, Toulouse cedex, France

    Alain Poli

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Carlet, C. (2006). On Bent and Highly Nonlinear Balanced/Resilient Functions and Their Algebraic Immunities. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2006. Lecture Notes in Computer Science, vol 3857. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11617983_1

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  • DOI: https://doi.org/10.1007/11617983_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-31423-3

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