The Relationship between Multiplicative Complexity and Nonlinearity

  • Joan Boyar
  • Magnus Gausdal Find
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8635)


We consider the relationship between nonlinearity and multiplicative complexity for Boolean functions with multiple outputs, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For quadratic circuits, we show that there is a tight connection between error correcting codes and circuits computing functions with high nonlinearity. Using known coding theory results, the lower bound proven here, for quadratic circuits for functions with n inputs and n outputs and high nonlinearity, shows that at least 2.32n AND gates are necessary. We further show that one cannot prove stronger lower bounds by only appealing to the nonlinearity of a function; we show a bilinear circuit computing a function with almost optimal nonlinearity with the number of AND gates being exactly the length of such a shortest code. For general circuits, we exhibit a concrete function with multiplicative complexity at least 2n − 3.


Boolean Function Linear Code Advance Encryption Standard General Circuit Bend Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Joan Boyar
    • 1
  • Magnus Gausdal Find
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkDenmark

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