Abstract
The performance of elliptic curve cryptosystems is primarily determined by an efficient implementation of the arithmetic operations in the underlying finite field. This paper presents a hardware architecture for a unified multiplier which operates in two types of finite fields: GF(p) and GF(2m). In both cases, the multiplication of field elements is performed by accumulation of partial-products to an intermediate result according to an MSB-first shift-and-add method. The reduction modulo the prime p (or the irreducible polynomial p(t), respectively) is interleaved with the addition steps by repeated subtractions of 2p and/or p (or p(t), respectively). A bit-serial multiplier executes a multiplication in GF(p) in approximately 1.5-[log2(p)] clock cycles, and the multiplication in GF(2m) takes exactly m clock cycles. The unified multiplier requires only slightly more area than that of the multiplier for prime fields GF(p). Moreover, it is shown that the proposed architecture is highly regular and simple to design.
Chapter PDF
Similar content being viewed by others
Keywords
References
P. Barrett. Implementing the Rivest, Shamir and Adleman public-key encryption algorithm on a standard digital signal processor. In A. M. Odlyzko (ed.), Advances in Cryptology — CRYPTO’ 86, vol. 263 of Lecture Notes in Computer Science, pp. 311–323. Springer-Verlag, Berlin, Germany, 1987.
T. Beth, B. M. Cook, and D. Gollmann. Architectures for exponentiation in GF(2n). In A. M. Odlyzko, (ed.), Advances in Cryptology — CRYPTO’ 86, vol. 263 of Lecture Notes in Computer Science, pp. 302–310. Springer-Verlag, Berlin, Germany, 1987.
I. F. Blake, G. Seroussi, and N. P. Smart. Elliptic Curves in Cryptography, vol. 265 of London Mathematial Society Lecture Notes Series. Cambridge University Press, Cambridge, UK, 1999.
G. R. Blakley. A computer algorithm for calculating the product AB modulo M. IEEE Transactions on Computers, 32(5):497–500, May 1983.
E. F. Brickell. A fast modular multiplication algorithm with application to two key cryptography. In D. Chaum, R. L. Rivest, and A. T. Sherman (eds.), Advances in Cryptology: Proceedings of CRYPTO’ 82, pp. 51–60. Plenum Press, New York, NY, USA, 1982.
J. Goodman and A. Chandrakasan. An energy efficient reconfigurable publik-key cryptography processor architecture. In Ç. K. Koç and C. Paar (eds.), Cryptographic Hardware and Embedded Systems — CHES 2000, vol. 1965 of Lecture Notes in Computer Science, pp. 174–191. Springer-Verlag, Berlin, Germany, 2000.
J. Groσchädl. A low-power bit-serial multiplier for finite fields GF(2m). In Proceedings of the 34th IEEE International Symposium on Circuits and Systems (ISCAS 2001), vol. IV, pp. 37–40, 2001.
Y.-J. Jeong and W. P. Burleson. VLSI array algorithms and architectures for RSA modular multiplication. IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 5(2):211–217, June 1997.
N. Koblitz. Elliptic curve cryptosystems. Mathematics of Computation, vol. 48, no. 177, pp. 203–209, January 1987.
N. Koblitz, A. J. Menezes, and S. A. Vanstone. The state of elliptic curve cryptography. Designs, Codes and Cryptography, 19(2/3):173–193, March 2000.
Ç. K. Koç and T. Acar. Montgomery multiplication in GF(2k). Designs, Codes and Cryptography, 14(1):57–69, April 1998.
P. Kornerup. High-radix modular multiplication for cryptosystems. In G. Jullien, M. J. Irwin, and E. E. Swartzlander (eds.), Proceedings of the 11th IEEE Symposium on Computer Arithmetic, pp. 277–283. IEEE Computer Society Press, Los Alamitos, CA, USA, 1993.
A. K. Lenstra and E. R. Verheul. Selecting cryptographic key sizes. In H. Imai and Y. Zheng (eds.), Public Key Cryptography — PKC 2000, vol. 1751 of Lecture Notes in Computer Science, pp. 446–465. Springer-Verlag, Berlin, Germany, 2000.
R. Lidl and H. Niederreiter. Introduction to Finite Fields and Their Applications. Second edition. Cambridge University Press, Cambridge, UK, 1994.
A. J. Menezes. Elliptic Curve Public Key Cryptosystems, vol. 234 of The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, Boston, MA, USA, 1993.
V. S. Miller. Use of elliptic curves in cryptography. In H. C. Williams (ed.), Advances in Cryptology — CRYPTO’ 85, vol. 218 of Lecture Notes in Computer Science, pp. 417–426. Springer-Verlag, Berlin, Germany, 1986.
P. L. Montgomery. Modular multiplication without trial division. Mathematics of Computation, 44(170):519–521, April 1985.
H. Morita. A fast modular-multiplication algorithm based on a higher radix. In G. Brassard (ed.), Advances in Cryptology — CRYPTO’ 89, vol. 435 of Lecture Notes in Computer Science, pp. 387–399. Springer-Verlag, Berlin, Germany, 1990.
National Institute of Standards and Technology (NIST). Digital Signature Standard (DSS). Federal Information Processing Standards (FIPS) Publication 186-2. Online available at http://csrc.nist.gov/encryption. February 2000.
H. Orup, E. Svendsen, and E. Andreasen. VICTOR an efficient RSA hardware implementation. In I. Damgård, (ed.), Advances in Cryptology — EUROCRYPT’ 90, vol. 473 of Lecture Notes in Computer Science, pp. 245–252. Springer-Verlag, Berlin, Germany, 1991.
B. Parhami. Computer Arithmetic: Algorithms and Hardware Designs. Oxford University Press, New York, NY, USA, 2000.
W. W. Peterson and E. J. Weldon. Error-Correcting Codes. Second edition. MIT Press, Cambridge, MA, USA, 1972.
E. Savaş, A. F. Tenca, and Ç. K. Koç. A scalable and unified multiplier architecture for finite fields GF(p) and GF(2m). In Ç. K. Koç and C. Paar (eds.), Cryptographic Hardware and Embedded Systems — CHES 2000, vol. 1965 of Lecture Notes in Computer Science, pp. 277–292. Springer-Verlag, Berlin, Germany, 2000.
H. Sedlak. The RSA cryptography processor. In D. Chaum and W. L. Price (eds.), Advances in Cryptology — EUROCRYPT’ 87, vol. 304 of Lecture Notes in Computer Science, pp. 95–105. Springer Verlag, Berlin, Germany, 1988.
N. Takagi. A radix-4 modular multiplication hardware algorithm for modular exponentiation. IEEE Transactions on Computers, 41(8):949–956, August 1992.
C. D. Walter. Faster modular multiplication by operand scaling. In J. Feigenbaum (ed.), Advances in Cryptology — CRYPTO’ 91, vol. 576 of Lecture Notes in Computer Science, pp. 313–323. Springer-Verlag, Berlin, Germany, 1992.
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
Großschädl, J. (2001). A Bit-Serial Unified Multiplier Architecture for Finite Fields GF(p) and GF(2m). In: Koç, Ç.K., Naccache, D., Paar, C. (eds) Cryptographic Hardware and Embedded Systems — CHES 2001. CHES 2001. Lecture Notes in Computer Science, vol 2162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44709-1_18
Download citation
DOI: https://doi.org/10.1007/3-540-44709-1_18
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42521-2
Online ISBN: 978-3-540-44709-2
eBook Packages: Springer Book Archive