## Introduction

Finite fields are the most commonly used arithmetical structures in cryptography [14,16] and coding [3,19,21]. Many algorithms in cryptographic and coding applications are defined in terms of finite field arithmetic operations. The elliptic curve cryptosystems [17,11] and the Diffie-Hellman key exchange [8] algorithm are important examples of such cryptographic applications. Also, common error control codes such as Reed-Solomon and BCH codes are based on finite field theory [4,21].

An algebraic field consists of a set and two operations defined over this set. The real numbers, the rational numbers, and the complex numbers under addition and multiplication are examples of algebraic fields. In fact, algebraic fields are the generalization of these usual number systems as described below.

One of the field operations satisfies the general properties of the usual addition. For this operation, an identity element exists and each element has an inverse. This identity element is...

## References

- 1.D. V. Bailey and C. Paar. Efficient arithmetic in finite field extensions with application in elliptic curve cryptography.
*Journal of Cryptology*, 2000.Google Scholar - 2.P. Barrett. Implementing the Rivest Shamir and Adleman public-key encryption algorithm on a standard digital signal processor. In A. M. Odlyzko, editor,
*Advances in Cryptology–-CRYPTO 86, Proceedings*, Lecture Notes in Computer Science, vol. 263, pp. 311–323. Springer, Berlin, Germany, 1986.Google Scholar - 3.E. Berlekamp.
*Algebraic Coding Theory*. McGraw-Hill, New York, NY, 1968.zbMATHGoogle Scholar - 4.R. Blahut.
*Theory and Practice of Error Control Codes*. Addison-Wesley, Reading, MA, 1983.zbMATHGoogle Scholar - 5.A. Bosselaers, R. Govaerts, and J. Vandewalle. Comparison of three modular reduction functions. In
*Crypto ’93*, Lecture Notes in Computer Science, vol. 773, pp. 175–186, 1994.Google Scholar - 6.M. Brown, D. Hankerson, J. López, and A. Menezes. Software implementation of the NIST elliptic curves over prime fields.
*Topics in Cryptology – CT-RSA 2001*, Lecture Notes in Computer Science, vol. 2020, pp. 250–265, Springer, Berlin, Germany, 2001Google Scholar - 7.J. F. Dhem. Efficient modular reduction algorithm in \(\mathcal{F}_q[x]\) and its application to “left to right” modular multiplication in \(\mathcal{F}_2[x]\). In C. D. Walter, editor,
*Cryptographic Hardware and Embedded Systems – CHES 2003*, Lecture Notes in Computer Science, vol. 2779, pp. 203–213. Springer, Berlin, Germany, 2003.Google Scholar - 8.W. Diffie and M. E. Hellman. New directions in cryptography.
*IEEE Transactions on Information Theory*, 22:644–654, November 1976.zbMATHCrossRefMathSciNetGoogle Scholar - 9.IEEE P1363. Standard specifications for public-key cryptography.Google Scholar
- 10.D. E. Knuth.
*The Art of Computer Programming, Volume 2, Seminumerical Algorithms*. Addison-Wesley, Reading, MA, Third edition, 1998.Google Scholar - 11.N. Koblitz. Elliptic curve cryptosystems.
*Mathematics of Computation*, 48(177):203–209, January 1987.zbMATHCrossRefMathSciNetGoogle Scholar - 12.Ç. K. Koç and T. Acar. Montgomery multiplication in GF\((2^k)\).
*Designs, Codes and Cryptography*, 14(1):57–69, April 1998.zbMATHCrossRefMathSciNetGoogle Scholar - 13.Ç. K. Koç, T. Acar, and B. S. Kaliski Jr. Analyzing and comparing Montgomery multiplication algorithms.
*IEEE Micro*, 16(3):26–33, June 1996.CrossRefGoogle Scholar - 14.R. J. McEliece.
*Finite Fields for Computer Scientists and Engineers*. Kluwer Academic Publishers, Boston, MA, Second edition, 1989.Google Scholar - 15.A. Menezes, P. Van Oorschot, and S. Vanstone.
*Handbook of Applied Cryptography*. CRC Press, Boca Raton, FL, 1997.zbMATHGoogle Scholar - 16.A. J. Menezes, I. F. Blake, X. Gao, R. C. Mullen, S. A. Vanstone, and T. Yaghoobian.
*Applications of Finite Fields*. Kluwer Academic Publishers, Boston, MA, 1993.zbMATHGoogle Scholar - 17.V. Miller. Uses of elliptic curves in cryptography. In H. C. Williams, editor,
*Advances in Cryptology–-CRYPTO 85, Proceedings*, Lecture Notes in Computer Science, No. 218, pp. 417–426. Springer, Berlin, Germany, 1985.Google Scholar - 18.P. L. Montgomery. Modular multiplication without trial division.
*Mathematics of Computation*, 44(170):519–521, April 1985.zbMATHCrossRefMathSciNetGoogle Scholar - 19.W. W. Peterson and E. J. Weldon Jr.
*Error-Correcting Codes*. MIT Press, Cambridge, MA, 1972.zbMATHGoogle Scholar - 20.J. Solinas.
*Generalized Mersenne numbers*. Technical Report CORR 99-39, Dept. of C&O, University of Waterloo, 1999.Google Scholar - 21.S. B. Wicker and V. K. Bhargava, editors.
*Reed-Solomon Codes and Their Applications*. IEEE Press, New York, NY, 1994.zbMATHGoogle Scholar

## Copyright information

© Springer Science+Business Media, LLC 2009