Designs, Codes and Cryptography

, Volume 38, Issue 2, pp 279–295

A New Characterization of Semi-bent and Bent Functions on Finite Fields*


DOI: 10.1007/s10623-005-6345-x

Cite this article as:
Khoo, K., Gong, G. & Stinson, D.R. Des Codes Crypt (2006) 38: 279. doi:10.1007/s10623-005-6345-x


We present a new characterization of semi-bent and bent quadratic functions on finite fields. First, we determine when a GF(2)-linear combination of Gold functions Tr(x2i+1) is semi-bent over GF(2n), n odd, by a polynomial GCD computation. By analyzing this GCD condition, we provide simpler characterizations of semi-bent functions. For example, we deduce that all linear combinations of Gold functions give rise to semi-bent functions over GF(2p) when p belongs to a certain class of primes. Second, we generalize our results to fields GF(pn) where p is an odd prime and n is odd. In that case, we can determine whether a GF(p)-linear combination of Gold functions Tr(xpi+1) is (generalized) semi-bent or bent by a polynomial GCD computation. Similar to the binary case, simple characterizations of these p-ary semi-bent and bent functions are provided.


bent functionssemi-bent functionscyclic matrixcyclic codesfinite fieldsSophie-German princes

AMS Classification


Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Khoongming Khoo
    • 1
  • Guang Gong
    • 2
  • Douglas R. Stinson
    • 3
  1. 1.DSO National LaboratoriesSingapore
  2. 2.Department of Electrical and Computer EngineeringUniversity of WaterlooWaterloo, Ont.Canada
  3. 3.School of Computer ScienceUniversity of WaterlooWaterloo, Ont.Canada