Abstract
This contribution describes how an elliptic curve cryptosystem can be implemented on very low cost microprocessors with reasonable performance. We focus in this paper on the Intel 8051 family of microcontrollers popular in smart cards and other cost-sensitive devices. The implementation is based on the use of the finite field GF((28 – 17)17) which is particularly suited for low end 8-bit processors. Two advantages of our method are that subfield modular reduction can be performed infrequently, and that an adaption of Itoh and Tsujii’s inversion algorithm is used for the group operation. We show that an elliptic curve scalar multiplication with a fixed point, which is the core operation for a signature generation, can be performed in a group of order approximately 2134 in less than 2 seconds Unlike other implementations, we do not make use of curves defined over a subfield such as Koblitz curves.
Keywords
References
Daniel V. Bailey. Optimal Extension Fields. Major Qualifying Project (Senior Thesis), 1998. Computer Science Department, Worcester Polytechnic Institute, Worcester, MA, USA.
Daniel V. Bailey and Christof Paar. Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms. In Advances in Cryptology - CRYPTO ‘88. Springer-Verlag Lecture Notes in Computer Science, 1998.
Daniel V. Bailey and Christof Paar. Efficient Arithmetic in Finite Field Extensions with Application in Elliptic Curve Cryptography. Journal of Cryptology, to appear.
I. Blake, G. Seroussi, and N. Smart. Elliptic Curves in Cryptography. Cambridge University Press, 1999.
E. F. Brickell, D. M. Gordon, K. S. McCurley, and D. B. Wilson. Fast exponentiation with precomputation. In Advances in Cryptography — EUROCRYPT ‘82, pages 200–207. Springer-Verlag, 1993.
Certicom Corp. The Elliptic Curve Cryptosystem for Smart Cards. online white paper, http://www.certicom.ca/ecc/wecc4.htm, 1998.
Peter de Rooij. Efficient exponentiation using precomputation and vector addition chains. In Advances in Cryptography — EUROCRYPT ‘88, pages 389–399. Springer-Verlag, 1998.
E. De Win, A. Bosselaers, S. Vandenberghe, P. De Gersem, and J. Vandewalle. A fast software implementation for arithmetic operations in GF(2m). In Asiacrypt ‘86. Springer-Verlag Lecture Notes in Computer Science, 1996.
E. De Win, S. Mister, B. Preneel, and M. Wiener. On the Performance of Signature Schemes Based on Elliptic Curves. In Algorithmic Number Theory: Third International Symposium, pages 252–266, Berlin, 1998. Springer-Verlag Lecture Notes in Computer Science.
P. Gaudry, F. Hess, and N. P. Smart. Constructive and Destructive Facets of Weil Descent on Elliptic Curves. technical report HPL 2000–10, http://www.hpl.hp.com/techreports/2000/HPL-2000–10.html, 2000.
Jorge Guajardo and Christof Paar. Efficient Algorithms for Elliptic Curve Cryptosystems. In Advances in Cryptology — Crypto ‘87, pages 342–356. Springer-Verlag Lecture Notes in Computer Science, August 1997.
R. Harley, D. Doligez, D. de Rauglaudre, and X. Leroy. http://cristal.inria.fr/%7Eharley/ecd17/.
IEEE. Standard Specifications for Public Key Cryptography. Draft, IEEE P1363 Standard, 1999. working document.
T. Itoh and S. Tsujii. A fast algorithm for computing multiplicative inverses in GF(2m) using normal bases. Information and Computation, 78: 171–177, 1988.
D. E. Knuth. The Art of Computer Programming. Volume 2: Seminumerical Algorithms. Addison-Wesley, Reading, Massachusetts, 2nd edition, 1981.
Tetsutaro Kobayashi, Hikaru Morita, Kunio Kobayashi, and Fumitaka Hoshino. Fast Elliptic Curve Algorithm Combining Frobenius Map and Table Reference to Adapt to Higher Characteristic. In Advances in Cryptography — EUROCRYPT’99. Springer-Verlag Lecture Notes in Computer Science, 1999.
Arjen Lenstra and Eric Verheul. Selecting cryptographic key sizes. In Public Key Cryptography — PKC 2000. Springer-Verlag Lecture Notes in Computer Science, 2000.
A. J. Menezes. Elliptic Curve Public Key Cryptosystems. Kluwer Academic Publishers, 1993.
A. J. Menezes, P. C. van Oorschot, and S. A. Vanstone. Handbook of Applied Cryptography. CRC Press, 1997.
P. Mihäilescu. Optimal Galois field bases which are not normal. Fast Software Encryption rump session, 1997.
D. Naccache and D. M’Raïhi. Cryptographic smart cards. IEEE Micro, 16 (3): 14–24, 1996.
D. Naccache, D. M’Raïhi, W. Wolfowicz, and A. di Porto. Are crypto-accelerators really inevitable? In Advances in Cryptography — EUROCRYPT ‘85, pages 404–409. Springer-Verlag Lecture Notes in Computer Science, 1995.
C. P. Schnorr. Efficient signature generation by smart cards. Journal of Cryptology, 4 (3): 161–174, 1991.
R. Schroeppel, H. Orman, S. O’Malley, and O. Spatscheck. Fast key exchange with elliptic curve systems. Advances in Cryptology — CRYPTO ‘85, pages 43–56, 1995.
Sencer Yeralan and Ashutosh Ahluwalia. Programming and Interfacing the 8051 Microcontroller. Addison-Wesley Publishing Company, 1995.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 IFIP International Federation for Information Processing
About this chapter
Cite this chapter
Woodbury, A.D., Bailey, D.V., Paar, C. (2000). Elliptic Curve Cryptography on Smart Cards without Coprocessors. In: Domingo-Ferrer, J., Chan, D., Watson, A. (eds) Smart Card Research and Advanced Applications. IFIP — The International Federation for Information Processing, vol 52. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-35528-3_5
Download citation
DOI: https://doi.org/10.1007/978-0-387-35528-3_5
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4757-6526-7
Online ISBN: 978-0-387-35528-3
eBook Packages: Springer Book Archive