Two efficient algorithms for arithmetic of elliptic curves using Frobenius map (original) (raw)

Abstract

In this paper, we present two efficient algorithms computing scalar multiplications of a point in an elliptic curve defined over a small finite field, the Frobenius map of which has small trace. Both methods use the identity which expresses multiplication-by-m maps by polynomials of Frobenius maps. Both are applicable for a large family of elliptic curves and more efficient than any other methods applicable for the family. More precisely, by Algorithm 1(Frobenius k_-ary method), we can compute mP in at most 2_l/5 + 28 elliptic additions for arbitrary l bit integer m and a point P on some elliptic curves. For other curves, the number of elliptic additions required is less than l. Algorithm 2(window method) requires at average 2_l_/3 elliptic additions to compute mP for l bit integer m and a point P on a family of elliptic curves. For some ‘good’ elliptic curves, it requires 5_l_/12 + 11 elliptic additions at average.

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Authors and Affiliations

  1. Electronics and Telecommunications Research Institute, 161 Kajong-Dong, Yusong-Gu, 305-350, Taejon, ROK
    Jung Hee Cheon, Sungmo Park, Sangwoo Park & Daeho Kim

Authors

  1. Jung Hee Cheon
  2. Sungmo Park
  3. Sangwoo Park
  4. Daeho Kim

Editor information

Hideki Imai Yuliang Zheng

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© 1998 Springer-Verlag Berlin Heidelberg

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Cheon, J.H., Park, S., Park, S., Kim, D. (1998). Two efficient algorithms for arithmetic of elliptic curves using Frobenius map. In: Imai, H., Zheng, Y. (eds) Public Key Cryptography. PKC 1998. Lecture Notes in Computer Science, vol 1431. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0054025

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