Distribution of Boolean Functions According to the Second-Order Nonlinearity

  • Stéphanie Dib
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6087)


The nonlinearity of a Boolean function is the minimum number of substitutions required in its truth table to change it into an affine function. Hence, in a cryptographic context, it is used to measure the strength of cryptosystems when facing linear attacks. As for the nonlinearity of order r of a Boolean function, which equals the least number of substitutions needed to change it into a function of degree at most r, it is examined when dealing with low-degree approximation attacks [7,14].

Many studies aimed at the distribution of Boolean functions according to the r-th order nonlinearity. Asymptotically, a lower bound is established in the higher order cases for almost all boolean functions, whereas a concentration point is shown in the (first order) nonlinearity case. We present a more accurate distribution by proving a concentration point in the second-order nonlinearity case.


Boolean functions nonlinearity Reed-Muller code 


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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Stéphanie Dib
    • 1
  1. 1.Institut de Mathématiques de LuminyMarseilleFrance

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