Acta Mathematica Hungarica

, Volume 140, Issue 1–2, pp 90–104 | Cite as

Existence and optimality of w-non-adjacent forms with an algebraic integer base

  • Clemens Heuberger
  • Daniel Krenn


We consider digit expansions in lattices with endomorphisms acting as base. We focus on the w-non-adjacent form (w-NAF), where each block of w consecutive digits contains at most one non-zero digit. We prove that for sufficiently large w and an expanding endomorphism, there is a suitable digit set such that each lattice element has an expansion as a w-NAF.

If the eigenvalues of the endomorphism are large enough and w is sufficiently large, then the w-NAF is shown to minimise the weight among all possible expansions of the same lattice element using the same digit system.

Key words and phrases

τ-adic expansion w-non-adjacent form redundant digit set lattice existence hyperelliptic curve cryptography Koblitz curve Frobenius endomorphism scalar multiplication Hamming weight optimality minimal expansion 

Mathematics Subject Classification

11A63 11H06 11R04 94A60 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2013

Authors and Affiliations

  1. 1.Institute of MathematicsAlpen-Adria-Universität KlagenfurtKlagenfurt am WörtherseeAustria
  2. 2.Institute of Optimisation and Discrete Mathematics (Math B)Graz University of TechnologyGrazAustria

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