Highly Nonlinear Balanced Boolean Functions with a Good Correlation-Immunity

  title={Highly Nonlinear Balanced Boolean Functions with a Good Correlation-Immunity},
  author={Eric Filiol and Caroline Fontaine},
We study a corpus of particular Boolean functions: the idempotents. They enable us to construct functions which achieve the best possible tradeoffs between the cryptographic fundamental properties: balancedness, correlation-immunity, a high degree and a high nonlinearity (that is a high distance from the affine functions). They all represent extremely secure cryptographic primitives to be implemented in stream ciphers. 
Further Results on the Relation Between Nonlinearity and Resiliency for Boolean Functions
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Further constructions of resilient Boolean functions with very high nonlinearity
A design of good combining Boolean functions that provide resilient Boolean functions with currently best known nonlinearity and achieve the provable upper bound on non linearity for resilience Boolean functions.
A Brief Outline of Research on Correlation Immune Functions
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This paper outlines the development in the field of constructing correlation immune Boolean functions with high nonlinearity and algebraic degree and briefly survey related issues in this area.
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Boolean functions with all main cryptographic properties
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It is proved that highly nonlinear functions usually have good propagation properties regarding different criteria and that most highly non linear functions with a three-valued Walsh spectrum can be transformed into 1-resilient functions.
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An operational reconstruction technique of most stream ciphers is presented, primarily exposed for key-stream generators which consist of several linear feedback shift registers combined by a nonlinear Boolean function.
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These properties of Boolean functions, including algebraic degree, linear structure property, and higher-order nonlinearity, are extensively studied in this chapter.


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A new algorithm is developed for making attacks to certain comparatively simple LFSR based ciphersystems and is applied to crack the random bit generators of Geffe and Beth-Piper.
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A new class of combining functions is presented, which provides better security against correlation attacks, and the security is quantified by the smallest number m + 1 of subsequences that must be simultaneously considered in a correlation attack.
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Nonlinearity criteria for Boolean functions are classified in view of their suitability for cryptographic design and two criteria turn out to be of special interest, the distance to linear structures and the Distance to affine functions, which are shown to be invariant under all affine transformations.
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A novel fast algorithm for the correlation attack on a class of stream ciphers is proposed. The algorithm is based on the error correction principle and the finite-state matrix representation of a
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The conclusion from the analysis is that the pseudonoise generator's output sequence and the sequences generated by the linear feedback shift registers should be uncorrelated, which leads to constraints for the nonlinear combining function to be used.
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A recursive definition of any correlation-immune function of maximal degree is given and the set of quadratic balanced correlation- immune functions of maximal order is described.
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