Abstract
This paper presents a new sliding window algorithm that is well-suited to an elliptic curve defined over an extension field for which the Frobenius map can be computed quickly, e.g., optimal extension field. The algorithm reduces elliptic curve group operations by approximately 15% for scalar multiplications for a practically used curve in comparison with Lim-Hwang's results presented at PKC2000, the fastest previously reported. The algorithm was implemented on computers. As a result, scalar multiplication can be accomplished in 573μs, 595μs, and 254μs on Pentium II (450 MHz), 21164A (500 MHz), and 21264 (500 MHz) computers, respectively.
This work was done while the author was in NTT Information Sharing Platform Laboratories.
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References
Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of applied cryptography. CRC Press (1997)
Bailey, D.V., Paar, C.: Optimal extension fields for fast arithmetic in public-key algorithms. In Krawczyk, H., ed.: Advances in Cryptology—CRYPTO’98. Volume 1462 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (1998) 472–485
Koblitz, N.: CM-curves with good cryptographic properties. In Feigenbaum, J., ed.: Advances in Cryptology — CRYPTO’91. Volume 576 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (1992) 279–287
Kobayashi, T., Morita, H., Kobayashi, K., Hoshino, F.: Fast elliptic curve algorithm combining frobenius map and table reference to adapt to higher characteristic. In Stern, J., ed.: Advances in Cryptology — EUROCRYPT’99. Volume 1592 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (1999) 176–189 (A preliminary version was written in Japanese and ented at SCIS’99-W4-1.4).
Solinas, J.A.: An improved algorithm for arithmetic on a family of elliptic curves. In Kaliski Jr., B.S., ed.: Advances in Cryptology — CRYPTO’97. Volume 1294 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (1997) 357–371
Lim, C.H., Lee, P.J.: More flexible exponentiation with precomputation. In Desmedt, Y.G., ed.: Advances in Cryptology — CRYPTO’94. Volume 839 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (1994) 95–107
Lim, C.H., Hwang, H.S.: Fast implementation of elliptic curve arithmetic in GF(pn). In Imai, H., Zheng, Y., eds.: Public Key Cryptography — Third International Workshop on Practice and Theory in Public Key Cryptosystems, PKC 2000. Volume 1751 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (2000) 405–421
Lim, C.H., Hwang, H.S.: Speeding up elliptic scalar multiplication with precomputation. In Song, J.S., ed.: Information Security and Cryptology — ICISC’99. Volume 1787 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (2000) 102–119
Tsuruoka, Y., Koyama, K.: Fast computation over elliptic curves E(F q n) based on optimal addition sequences. IEICE Transactions Fundamentals of Electronics, Communications and Computer Sciences (Japan) E84-A (2001) 114–119
Cohen, H., Miyaji, A., Ono, T.: Efficient elliptic curve exponentiation using mixed coordinates. In Ohta, K., Pei, D., eds.: Advances in Cryptology—ASIACRYPT’98. Volume 1514 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (1998) 51–65
Aoki, K., Hoshino, F., Kobayashi, T., Oguro, H.: Elliptic curve arithmetic using SIMD. In Davida, G., Frankel, Y., eds.: Information Security Conference—ISC’01. Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York (2001) to appear. (Preliminary version written in Japanese was appeared in SCIS2000-B05 and ISEC2000-161.).
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Aoki, K., Hoshino, F., Kobayashi, T. (2001). A Cyclic Window Algorithm for ECC Defined over Extension Fields. In: Qing, S., Okamoto, T., Zhou, J. (eds) Information and Communications Security. ICICS 2001. Lecture Notes in Computer Science, vol 2229. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45600-7_8
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DOI: https://doi.org/10.1007/3-540-45600-7_8
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