Abstract
This chapter discusses building blocks for implementing popular public key cryptosystems, like RSA, Diffie-Hellman Key Exchange (DHKE) and Elliptic Curve Cryptography (ECC). Therefore, we briefly introduce field-based arithmetic on which most of recently established public key cryptosystems rely. As most popular fields, we give examples for architecture implementing efficient arithmetic operations over prime and binary extension fields for use in cryptographic applications.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
According to [14], the discovery of public-key cryptography (PKC) in the intelligence community is attributed to John H. Ellis in 1970. The discovery of the equivalent of the RSA cryptosystem [38] is attributed to Clifford Cocks in 1973 while the equivalent of the Diffie–Hellman key exchange was discovered by Malcolm J. Williamson, in 1974. However, it is believed that these British scientists did not realize the practical implications of their discoveries at the time of their publication (see, for example, [39, 11]).
- 2.
It is important to understand that NP-complete problems from computer science cannot be simply converted for use with PKC since a one-way function for PKC must guarantee hardness in all cases which is usually not the case for all NP-complete problems.
- 3.
Recall that the time complexity of n-bit addition or subtraction is only in \(\mathcal{O}(n)\).
- 4.
References
FIPS 186-2: Digital Signature Standard (DSS). 186-2, February 2000. Available for download at http://csrc.nist.gov/encryption.
D. N. Amanor, C. Paar, J. Pelzl, V. Bunimov, and M. Schimmler. Efficient Hardware Architectures for Modular Multiplication on FPGAs. In 2005 International Conference on Field Programmable Logic and Applications (FPL), Tampere, Finland, pages 539–542. IEEE Circuits and Systems Society, August 2005.
D. V. Bailey and C. Paar. Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms. In H. Krawczyk, editor, Advances in Cryptology — CRYPTO ’98, volume LNCS 1462, pages 472–485, Springer-Verlag, Berlin, 1998.
D. V. Bailey and C. Paar. Efficient Arithmetic in Finite Field Extensions with Application in Elliptic Curve Cryptography. Journal of Cryptology, 14(3):153–176, 2001.
P. Barrett. Implementing the Rivest, Shamir and Adleman public-key encryption algorithm on standard digital signal processor. In A. Odlyzko, editor, Advances in Cryptology — CRYPTO’86, volume 263 of LNCS, pages 311–323. Springer-Verlag, Berlin 1987.
L. Batina, S. B. Ors, B. Preneel, and J. Vandewalle. Hardware architectures for public key cryptography. Integration, the VLSI Journal, 34(6):1–64, 2003.
G. Blakley. A computer algorithm for calculating the product \(A \cdot B\) modulo M. IEEE Transactions on Computers, C-32(5):497–500, May 1983.
D. Boneh and M. Franklin. Identity-Based Encryption from the Weil Pairing. In J. Kilian, editor, Advances in Cryptology — CRYPTO 2001, volume LNCS 2139, pages 213–229. Springer-Verlag, Berlin 2001.
V. Bunimov and M. Schimmler. Area and Time Efficient Modular Multiplication of Large Integers. In IEEE 14th International Conference on Application-specific Systems, Architectures and Processors, June 2003.
A. Daly, L. Marnaney, and E. Popovici. Fast Modular Inversion in the Montgomery Domain on Reconfigurable Logic. Technical report, University College Cork, Cork, Ireland, 2004.
W. Diffie. Subject: Authenticity of Non-secret Encryption documents. World Wide Web, October 6, 1999. Email message sent to John Young. Available at http://cryptome.org/ukpk-diffie.htm.
W. Diffie and M. E. Hellman. New directions in cryptography. IEEE Transactions on Information Theory, IT-22(6):644–654, November 1976.
T. ElGamal. A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Transactions on Information Theory, 31:469–472, 1985.
J. H. Ellis. The Story of Non-secret Encryption. Available at http://jya.com/ellisdoc.htm, December 16th, 1997.
I. E. T. Force. The Kerberos Network Authentication Service (V5). RFC 4120, July 2005.
D. M. Gordon. A survey of fast exponentiation methods. Journal of Algorithms, 27:129–146, 1998.
J. Guajardo, T. Güneysu, S. S. Kumar, C. Paar, and J. Pelzl. Efficient hardware implementation of finite fields with applications to cryptography. Acta Applicandae Mathematicae, 93:75–118, 2006.
J. Guajardo and C. Paar. Efficient Algorithms for Elliptic Curve Cryptosystems. In B. Kaliski, Jr., editor, Advances in Cryptology — CRYPTO ’97, volume 1294, pages 342–356, Springer Verlag, Berlin August 1997.
J. Hoffstein, D. Lieman, J. Pipher, and J. H. Silverman. NTRU: A Public Key Cryptosystem. Technical report, Aug. 11 1999.
K. Hwang. Computer Arithmetic: Principles, Architecture and Design. John Wiley & Sons, Inc. New York, 1979.
T. Itoh and S. Tsujii. A fast algorithm for computing multiplicative inverses in \(GF(2^m)\) using normal bases. Information and Computation, 78:171–177, 1988.
D. Knuth. The Art of Computer Programming, Seminumerical Algorithms, volume 2. Addison-Wesley, Reading, MA November 1971. 2nd printing.
D. E. Knuth. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, volume 2. Second edition, Addison-Wesley, Reading, MA 1973.
N. Koblitz. Elliptic curve cryptosystems. Mathematics of Computation, 48(177):203–209, January 1987.
N. Koblitz. Hyperelliptic cryptosystems. Journal of Cryptology, 1(3):129–150, 1989.
N. Koblitz. A Course in Number Theory and Cryptography. Springer Verlag, New York, 1994.
N. Koblitz. An Elliptic Curve Implementation of the Finite Field Digital Signature Algorithm. In H. Krawczyk, editor, Advances in Cryptology — CRYPTO 98, volume LNCS 1462, pages 327–337. Springer-Verlag, Berlin 1998.
Ç. K. Koç, T. Acar, and B. S. Kaliski. Analyzing and comparing Montgomery multiplication algorithms. IEEE Micro, 16(3):26–33, June 1996.
A. Lenstra and E. Verheul. Selecting Cryptographic Key Sizes. In H. Imai and Y. Zheng, editors, Practice and Theory in Public Key Cryptography–-PKC 2000, volume 1751, pages 446–465, January 2000.
R. J. McEliece. A public-key cryptosystem based on algebraic coding theory. DSN Progress Report, pages 42–44, 1987.
A. J. Menezes, P. C. van Oorschot, and S. A. Vanstone. Handbook of Applied Cryptography. The CRC Press series on discrete mathematics and its applications. 1997.
R. C. Merkle. Secure communications over insecure channels. Communications of the ACM, 21(4):294–299, 1978.
P. Mihăilescu. Optimal Galois Field Bases Which Are Not Normal. Recent Results Session — FSE ’97, 1997.
V. S. Miller. Use of Elliptic Curves in Cryptography. In H. C. Williams, editor, Advances in Cryptology — CRYPTO ’85, volume 218, pages 417–426, August 1986.
P. Montgomery. Modular multiplication Without trial division. Mathematics of Computation, 44(170):519–521, April 1985.
National Institute of Standards and Technology (NIST). Recommended Elliptic Curves for Federal Government Use, July 1999. csrc.nist.gov/csrc/fedstandards.html.
J. Pollard. Monte Carlo methods for index computation mod p. Mathematics of Computation, 32(143):918–924, July 1978.
R. L. Rivest, A. Shamir, and L. Adleman. A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM, 21(2):120–126, February 1978.
B. Schneier. Crypto-Gram Newsletter. World Wide Web, May 15, 1998. Available at http://www.schneier.com/crypto-gram-9805.html.
K. Sloan. Comments on a computer algorithm for calculating the product \(A \cdot B\) modulo M. IEEE Transactions on Computers, C-34(3):290–292, March 1985.
N. Smart. Elliptic curve cryptosystems over small fields of odd characteristic. Journal of Cryptology, 12(2):141–151, Spring 1999.
J. Solinas. Generalized Mersenne Numbers. Technical Report, CORR 99-39, Department of Combinatorics and Optimization, University of Waterloo, Canada,, 1999.
L. Song and K. K. Parhi. Low energy digit-serial/parallel finite field multipliers. Journal of VLSI Signal Processing, 19(2):149–166, June 1998.
J. von zur Gathen and M. Nöcker. Exponentiation in Finite Fields: Theory and Practice. In T. Mora and H. Mattson, editors, Applied Algebra, Algebraic Algorithms and Error Correcting Codes — AAECC-12, volume LNCS 1255, pages 88–113. Springer-Verlag, 2000.
C. Walter. Logarithmic speed modular multiplication. Electronics Letters, 30(17):1397–1398, 1994.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Güneysu, T., Paar, C. (2010). Modular Integer Arithmetic for Public Key Cryptography. In: Verbauwhede, I. (eds) Secure Integrated Circuits and Systems. Integrated Circuits and Systems. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-71829-3_1
Download citation
DOI: https://doi.org/10.1007/978-0-387-71829-3_1
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-71827-9
Online ISBN: 978-0-387-71829-3
eBook Packages: EngineeringEngineering (R0)