Generalised complementary arrays

  • Authors:

  • Matthew G. Parker;Constanza Riera

  • Affiliations:

  • The Selmer Center, Department of Informatics, University of Bergen, Bergen, Norway;Høgskolen i Bergen, Bergen, Norway

  • Venue:
  • IMACC'11 Proceedings of the 13th IMA international conference on Cryptography and Coding
  • Year:
  • 2011

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Abstract

We present a generalised setting for the construction of complementary array pairs and its proof, using a unitary matrix notation. When the unitaries comprise multivariate polynomials in complex space, we show that four definitions of conjugation imply four types of complementary pair - types I, II, III, and IV. We provide a construction for complementary pairs of types I, II, and III over {1,−1}, and further specialize to a construction for all known 2 ×2 ×…×2 complementary array pairs of types I, II, and III over {1,−1}. We present a construction for type-IV complementary array pairs, and call them Rayleigh quotient pairs . We then generalise to complementary array sets, provide a construction for complementary sets of types I, II, and III over {1,−1}, further specialize to a construction for all known 2 ×2 ×…×2 complementary array sets of types I, II, and III over {1,−1}, and derive closed-form Boolean formulas for these cases.