Abstract
We construct a supersingular implementation of the Elliptic Curve Digital Signature Algorithm (ECDSA) that is essentially equivalent to a finite field implementation of the Digital Signature Algorithm (DSA), and then we compare the efficiency of the two systems. The elliptic curve method is about 12 times faster. In the last section we use the same ideas to give a particularly efficient nonsupersingular implementation of elliptic curve cryptography in characteristic 7.
Chapter PDF
References
R. Balasubramanian and N. Koblitz, The improbability than an elliptic curve has subexponential discrete log problem under the Menezes-Okamoto-Vanstone algorithm, J. Cryptology 11 (1998), 141–145.
I. Blake, X. H. Gao, R. C. Mullin, S. A. Vanstone, and T. Yaghoobian, Applications of Finite Fields, Kluwer Acad. Publ., 1993.
S. Gao and H. W. Lenstra, Jr., Optimal normal bases, Designs, Codes and Cryptography 2 (1992), 315–323.
K. Ireland and M. I. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Springer-Verlag, 1990.
N. Koblitz, Elliptic curve cryptosystems, Math. Comp. 48 (1987), 203–209.
N. Koblitz, CM-curves with good cryptographic properties, Advances in Cryptology — Crypto '91, Springer-Verlag, 1992, 279–287.
N. Koblitz, A Course in Number Theory and Cryptography, 2nd ed., Springer-Verlag, 1994.
N. Koblitz, Algebraic Aspects of Cryptography, Springer-Verlag, 1998.
N. Koblitz, A. Menezes, and S. A. Vanstone, The state of elliptic curve cryptography, to appear in Designs, Codes and Cryptography.
W. Meier and O. Staffelbach, Efficient multiplication on certain non-supersingular elliptic curves, Advances in Cryptology — Crypto '92, Springer-Verlag, 1993, 333–344.
A. Menezes, Elliptic Curve Public Key Cryptosystems, Kluwer Acad. Publ., 1993.
A. Menezes, T. Okamoto, and S. A. Vanstone, Reducing elliptic curve logarithms to logarithms in a finite field, IEEE Trans. Information Theory 39 (1993), 1639–1646.
A. Menezes and S. A. Vanstone, Elliptic curve cryptosystems and their implementation, J. Cryptology 6 (1993), 209–224.
V. Miller, Uses of elliptic curves in cryptography, Advances in Cryptology — Crypto '85, Springer-Verlag, 1986, 417–426.
R. Mullin, I. Onyszchuk, S. A. Vanstone, and R. Wilson, Optimal normal bases in GF(p n), Discrete Applied Math. 22 (1988/89), 149–161.
National Institute for Standards and Technology, Digital signature standard, FIPS Publication 186, 1993.
T. Satoh and K. Araki, Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves, preprint.
R. Schroeppel, personal communication, Dec. 2, 1997.
R. Schroeppel, H. Orman, S. O'Malley, and O. Spatscheck, Fast key exchange with elliptic curve systems, Advances in Cryptology — Crypto '95, Springer-Verlag, 1995, 43–56.
I. A. Semaev, Evaluation of discrete logarithms in a group of p-torsion points of an elliptic curve in characteristic p, Math. Comp. 67 (1998), 353–356.
J. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1986.
N. Smart, The discrete log problem on elliptic curves of trace 1, preprint.
J. Solinas, An improved algorithm for arithmetic on a family of elliptic curves, Advances in Cryptology — Crypto '97, Springer-Verlag, 1997, 357–371.
E. De Win, A. Bosselaers, S. Vandenberghe, P. De Gersem, and J. Vandewalle, A fast software implementation for arithmetic operations in GF(2n), Advances in Cryptology — Asiacrypt '96, Springer-Verlag, 1996, 65–76.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1998 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Koblitz, N. (1998). An elliptic curve implementation of the finite field digital signature algorithm. In: Krawczyk, H. (eds) Advances in Cryptology — CRYPTO '98. CRYPTO 1998. Lecture Notes in Computer Science, vol 1462. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055739
Download citation
DOI: https://doi.org/10.1007/BFb0055739
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-64892-5
Online ISBN: 978-3-540-68462-6
eBook Packages: Springer Book Archive