Abstract
We discuss multidoubling methods for efficient elliptic scalar multiplication. The methods allows computation of 2k P directly from P without computing the intermediate points, where P denotes a randomly selected point on an elliptic curve. We introduce algorithms for elliptic curves with Montgomery form and Weierstrass form defined over finite fields with characteristic greater than 3 in terms of affine coordinates. These algorithms are faster than k repeated doublings. Moreover, we apply the algorithms to scalar multiplication on elliptic curves and analyze computational complexity. As a result of our implementation with respect to the Montgomery and Weierstrass forms in terms of affine coordinates, we achieved running time reduced by 28% and 31%, respectively, in the scalar multiplication of an elliptic curve of size 160-bit over finite fields with characteristic greater than 3.
Chapter PDF
Keywords
References
G. B. Agnew, R. C. Mullin, S. A. Vanstone, “An Implementation of Elliptic Curve Cryptosystems Over F 1552 ”, IEEE journal on selected areas in communications, 11, No.5 (1993)
H. Cohen, A. Miyaji, T. Ono, “Efficient Elliptic Curve Exponentiation Using Mixed Coordinates”, Advances in Cryptology—ASIACRYPT’98, LNCS, 1514 (1998), Springer-Verlag, 51–65.
D. M. Gordon, “A survey of fast exponentiation methods”, Journal of Algorithms, 27 (1998), 129–146.
J. Guajardo, C. Paar, “Efficient Algorithms for Elliptic Curve Cryptosystems”, Advances in Cryptology—CRYPTO’97, LNCS, 1294 (1997), Springer-Verlag, 342–356.
Y. Han, P. C. Tan, “Direct Computation for Elliptic Curve Cryptosystems”, Pre-proceedings of CHES’99, (1999), Springer-Verlag, 328–340.
K. Itoh, M. Takenaka, N. Torii, S. Temma, Y. Kurihara, “Fast Implementation of Public-key Cryptography on a DSP TMS320C6201”, Cryptography Hardware and Embedded Systems, LNCS, 1717 (1999), Springer-Verlag, 61–72.
T. Izu, “Elliptic Curve Exponentiation for Cryptosystem”, The 1999 Symposium on Cryptography and Information Security, (1999), 275–280.
N. Koblitz, “Elliptic curve cryptosystems”, Mathematics of Computation, 48 (1987), 203–209.
K. Koyama, Y. Tsuruoka, “Speeding up Elliptic Cryptosystems by Using a Signed Binary Window Method”, Advances in Cryptology—CRYPTO’92, LNCS, 740 (1993), Springer-Verlag, 345–357.
J. López, R. Dahab, “Fast Multiplication on Elliptic Curves over GF(2m) without Precomputation”, CHES’99, LNCS, 1717 (1999), Springer-Verlag, 316–327.
H. W. Lenstra, Jr., “Factoring integers with elliptic curves”, Ann. of Math, 126 (1987), 649–673.
V. Miller, “Uses of elliptic curves in cryptography”, Advances in Cryptology—CRYPTO’85, LNCS, 218 (1986), Springer-Verlag, 417–426.
A. Menezes, A. Vanstone, “Elliptic Curve Cryptosystems and Their Implementation”, J. Cryptology, 6 (1993), Springer-Verlag, 209–224.
P. L. Montgomery, “Speeding the Pollard and Elliptic Curve Methods of Factorization”, Mathematics of Computation, 48 (1987), 243–264.
A. Miyaji, T. Ono, H. Cohen, “Efficient Elliptic Curve Exponentiation (I)”, Technical Report of IEICE, ISEC97-16, (1997)
A. Miyaji, T. Ono, H. Cohen, “Efficient Elliptic Curve Exponentiation”, Advances in Cryptology—ICICS’97, LNCS, 1334 (1997), Springer-Verlag, 282–290.
V. Müller, “Efficient Algorithms for Multiplication on Elliptic Curves”, Proceedings of GI—Arbeitskonferenz Chipkarten 1998, TU München, (1998)
K. Okeya, H. Kurumatani, K. Sakurai, “Elliptic Curves with the Montgomery Form and Their Cryptographic Applications”, Public Key Cryptography (PKC) 2000, LNCS, 1751 (2000), Springer-Verlag, 238–257.
Y. Sakai, K. Sakurai, “Efficient Scalar Multiplications on Elliptic Curves without Repeated Doublings and Their Practical Performance”. Information Security and Privacy, ACISP 2000, LNCS, 1841 (2000), Springer-Verlag, 59–63. The final version of this paper has been published [SS01].
Y. Sakai, K. Sakurai, “Efficient Scalar Multiplications on Elliptic Curves with Direct Computations of Several Doublings”. IEICE Trans. Fundamentals, E84-A No.1 (2001), 120–129. Available at http://search.ieice.or.jp/2001/.les/e120a01.htm#e84-a,1,107
E. De Win, S. Mister, B. Preneel, M. Wiener, “On the Performance of Signature Schemes Based on Elliptic Curves”, Algorithmic Number Theory III, LNCS, 1423 (1998), Springer-Verlag, 252–266.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sakai, Y., Sakurai, K. (2001). On the Power of Multidoubling in Speeding Up Elliptic Scalar Multiplication. In: Vaudenay, S., Youssef, A.M. (eds) Selected Areas in Cryptography. SAC 2001. Lecture Notes in Computer Science, vol 2259. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45537-X_21
Download citation
DOI: https://doi.org/10.1007/3-540-45537-X_21
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43066-7
Online ISBN: 978-3-540-45537-0
eBook Packages: Springer Book Archive