On the Higher Order Nonlinearities of Boolean Functions and S-Boxes, and Their Generalizations

(Invited Paper)
  • Claude Carlet
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5203)

Abstract

The r-th order nonlinearity of a Boolean function \(f:F_2^n\to F_2\) is its minimum Hamming distance to all functions of algebraic degrees at most r, where r is a positive integer. The r-th order nonlinearity of an S-box \(F:F_2^n\to F_2^m\) is the minimum r-th order nonlinearity of its component functions v·F, \(v\in F_2^m\setminus \{0\}\). The role of this cryptographic criterion against attacks on stream and block ciphers has been illustrated by several papers. Its study is also interesting for coding theory and is related to the covering radius of Reed-Muller codes (i.e. the maximum multiplicity of errors that have to be corrected when maximum likelihood decoding is used on a binary symmetric channel). We give a survey of what is known on this parameter, including the bounds involving the algebraic immunity of the function, the bounds involving the higher order nonlinearities of its derivatives, and the resulting bounds on the higher order nonlinearities of the multiplicative inverse functions (used in the S-boxes of the AES). We show an improvement, when we consider an S-box instead of a Boolean function, of the bounds on the higher order nonlinearity expressed by means of the algebraic immunity. We study a generalization (for S-boxes) of the notion and we give new results on it.

Keywords

Block cipher Boolean function Covering radius Cryptography Higher-order nonlinearity Reed-Muller code S-box Stream cipher 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Claude Carlet
    • 1
  1. 1.Department of Mathematics (MAATICAH)University of Paris 8Saint-Denis CedexFrance

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