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Sequences, Bent Functions and Jacobsthal Sums

  • Tor Helleseth
  • Alexander Kholosha
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6338)

Abstract

The p-ary function f(x) mapping GF(p 4k ) to GF(p) and given by \(f(x)={\rm Tr}_{4k}\big(ax^d+bx^2\big)\) with a,b ∈ GF(p 4k ) and d = p 3k  + p 2k  − p k  + 1 is studied with the respect to its exponential sum. In the case when either \(a^{p^k(p^k+1)}\neq b^{p^k+1}\) or a 2 = b d with b ≠ 0, this sum is shown to be three-valued and the values are determined. For the remaining cases, the value of the exponential sum is expressed using Jacobsthal sums of order p k  + 1. Finding the values and the distribution of those sums is a long-lasting open problem.

Keywords

Cyclotomic number Jacobsthal sum p-ary bent function polynomial over finite field Walsh transform 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Tor Helleseth
    • 1
  • Alexander Kholosha
    • 1
  1. 1.The Selmer Center, Department of InformaticsUniversity of BergenBergenNorway

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