Abstract
In this work we analyse the GLV method of Gallant, Lambert and Vanstone (CRYPTO 2001) which uses a fast endomorphism Φ with minimal polynomial X 2 +rX +s to compute any multiple kP of a point P of order n lying on an elliptic curve. First we fill in a gap in the proof of the bound of the kernel K vectors of the reduction map f : (i, j)→ i+λj (mod n). In particular, we prove the GLV decomposition with explicit constant kP = k1P + k 2Φ(P), with max Rik 1∣, ∣k 2∣ ⪯ √1 +∣r∣ + s√n . Next we improve on this bound and give the best constant in the given examples for the quantity supk,n max ∣k 1∣, ∣k 2∣/√n. Independently Park, Jeong, Kim, and Lim (PKC 2002) have given similar but slightly weaker bounds. Finally we provide the first explicit bounds for the GLV method generalised to hyperelliptic curves as described in Park, Jeong and Lim (EUROCRYPT 2002).
The work described in this paper has been supported [in part] by the Commission of the European Communities through the IST Programme under Contract IST- 1999-12324, http://www.cryptonessie.org/. The information in this document is provided as is, and no guarantee or warranty is given or implied that the information is fit for any particular purpose. The user thereof uses the information at its sole risk and liability. The views expressed are those of the authors and do not represent an official view/position of the NESSIE project (as a whole).
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Sica, F., Ciet, M., Quisquater, JJ. (2003). Analysis of the Gallant-Lambert-Vanstone Method Based on Efficient Endomorphisms: Elliptic and Hyperelliptic Curves. In: Nyberg, K., Heys, H. (eds) Selected Areas in Cryptography. SAC 2002. Lecture Notes in Computer Science, vol 2595. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36492-7_3
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DOI: https://doi.org/10.1007/3-540-36492-7_3
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