Provably Sublinear Point Multiplication on Koblitz Curves and its Hardware Implementation

V. S.  Dimitrov; K. U.  Jaervinen; M. J.  Jacobson Jr.; W. F.  Chan; Z.  Huang

Paper 2006/305

Provably Sublinear Point Multiplication on Koblitz Curves and its Hardware Implementation

V. S. Dimitrov, K. U. Jaervinen, M. J. Jacobson Jr., W. F. Chan, and Z. Huang

Abstract

We describe algorithms for point multiplication on Koblitz curves using multiple-base expansions of the form $k = \sum \pm τ^{a} (τ - 1)^{b}$ and $k = \sum \pm τ^{a} (τ - 1)^{b} (τ^{2} - τ - 1)^{c} .$ We prove that the number of terms in the second type is sublinear in the bit length of k, which leads to the first provably sublinear point multiplication algorithm on Koblitz curves. For the first type, we conjecture that the number of terms is sublinear and provide numerical evidence demonstrating that the number of terms is significantly less than that of -adic non-adjacent form expansions. We present details of an innovative FPGA implementation of our algorithm and performance data demonstrating the efficiency of our method.

Metadata

Available format(s): PDF PS
Category: Public-key cryptography
Publication info: Published elsewhere. This is an extended version of our paper accepted to CHES 2006.
Keywords: elliptic curve cryptosystems Koblitz curves point multiplication double-base number systems hardware implementation
Contact author(s): jacobs @ cpsc ucalgary ca
History: 2006-09-07: revised; 2006-09-06: received; See all versions
Short URL: https://ia.cr/2006/305
License: CC BY

BibTeX

@misc{cryptoeprint:2006/305,
      author = {V. S.  Dimitrov and K. U.  Jaervinen and M. J.  Jacobson Jr. and W. F.  Chan and Z.  Huang},
      title = {Provably Sublinear Point Multiplication on Koblitz Curves and its Hardware Implementation},
      howpublished = {Cryptology {ePrint} Archive, Paper 2006/305},
      year = {2006},
      url = {https://eprint.iacr.org/2006/305}
}