Radix-r Non-Adjacent Form

  • Tsuyoshi Takagi
  • Sung-Ming Yen
  • Bo-Ching Wu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3225)

Abstract

Recently, the radix-3 representation of integers is used for the efficient implementation of pairing based cryptosystems. In this paper, we propose non-adjacent form of radix-r representation (rNAF) and efficient algorithms for generating rNAF. The number of non-trivial digits is (r–2)(r+1)/2 and its average density of non-zero digit is asymptotically (r–1)/(2r–1). For r=3, the non-trivial digits are { ± 2, ± 4} and the non-zero density is 0.4. We then investigate the width-w version of rNAF for the general radix-r representation, which is a natural extension of the width-w NAF. Finally we compare the proposed algorithms with the generalized NAF (gNAF) discussed by Joye and Yen. The proposed scheme requires a larger table but its non-zero density is smaller even for large radix. We explain that gNAF is a simple degeneration of rNAF — we can consider that rNAF is a canonical form for the radix-r representation. Therefore, rNAF is a good alternative to gNAF.

Keywords

Non-adjacent form radix-r representation signed window method elliptic curve cryptosystem pairing based cryptosystem 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Tsuyoshi Takagi
    • 1
  • Sung-Ming Yen
    • 2
  • Bo-Ching Wu
    • 2
  1. 1.Fachbereich InformatikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Laboratory of Cryptography and Information Security (LCIS), Dept of Computer Science and Information EngineeringNational Central UniversityChung-LiTaiwan, R.O.C.

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