Efficient Computation of the Best Quadratic Approximations of Cubic Boolean Functions

  • Nicholas Kolokotronis
  • Konstantinos Limniotis
  • Nicholas Kalouptsidis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4887)

Abstract

The problem of computing best quadratic approximations of a subset of cubic functions with arbitrary number of variables is treated in this paper. We provide methods for their efficient calculation by means of best affine approximations of quadratic functions, for which formulas for their direct computation, without using Walsh-Hadamard transform, are proved. The notion of second-order nonlinearity is introduced as the minimum distance from all quadratic functions. Cubic functions, in the above subset, with maximum second-order nonlinearity are determined, leading to a new lower bound for the covering radius of the second order Reed-Muller code \(\Re(2,n)\). Moreover, a preliminary study of the second-order nonlinearity of known bent functions constructions is also given.

Keywords

Boolean functions bent functions covering radius second order nonlinearity low-order approximations Reed-Muller codes 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Nicholas Kolokotronis
    • 1
    • 2
  • Konstantinos Limniotis
    • 1
  • Nicholas Kalouptsidis
    • 1
  1. 1.Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, TYPA Buildings, University Campus, 15784 AthensGreece
  2. 2.Department of Computer Science and Technology, University of Peloponnese, End of Karaiskaki Street, 22100 TripolisGreece

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