Efficient Unified Arithmetic for Hardware Cryptography

  • Erkay Savaş
  • Çetin Kaya Koç


The basic arithmetic operations (i.e., addition, multiplication, and inversion) in finite fields, \(GF(q)\)


Full Adder Elliptic Curve Cryptography Subtraction Operation Critical Path Delay Adder Circuit 
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.


  1. 1.
    W. Diffie and M. E. Hellman. New directions in cryptography. IEEE Transactions on Information Theory, 22:644–654, November 1976.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    National Institute for Standards and Technology. Digital Signature Standard (DSS), Federal Register, 56:169, August 1991.Google Scholar
  3. 3.
    N. Koblitz, Elliptic curve cryptosystems. Mathematics Computation, 48(177):203–209, January 1987.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    A. J. Menezes.Elliptic Curve Public Key ryptosystems. Kluwer Academic Publishers, Boston, MA, 1993.Google Scholar
  5. 5.
    D. Boneh and M. Franklin.Identity-based Encryption from the Weil Pairing.In Advances in Cryptology – CRYPTO 2001, volume 2139 of Lecture Notes in Computer Science, pp. 213–229. Springer-Verlag, 2001.Google Scholar
  6. 6.
    A. Shamir. Identity-Based Cryptosystems and Signature Schemes. In Advances in Cryptology - YPTO 1985, volume 196 of Lecture Notes in Computer ience,pp. 47–53. Springer-Verlag, 1985.Google Scholar
  7. .7
    P. L. Montgomery. Modular multiplication ithout trial division.Mathematics of Computation, 4(170):519–521, April 1985.CrossRefGoogle Scholar
  8. 8.
    Ç. K. Koç and T. Acar. Montgomery multiplication in GF\((2^k)\). In Proceedings of Third Annual Workshop on Selected Areas in Cryptography,pp. 95–106, Queen’s University, Kingston, Ontario, Canada, August 15–16 1996.Google Scholar
  9. 9.
    IEEE. P1363. Standard pecifications for public-key cryptography. 2000.Google Scholar
  10. 10.
    E. Savaş, A. F. Tenca, and Ç. Ko ç,A scalable and unified multiplier architecture for inite fields \(GF(p)\) and \(GF(2^m)\). In Cryptographic Hardware and Embedded Systems, Workshop on Cryptographic Hardware and Embedded Systems, pp. 277-292. Springer-Verlag, Berlin, 2000.Google Scholar
  11. 11.
    A. F. Tenca and Ç K. Koç.“A Scalable Architecture for Montgomery multiplication”,Lecture Notes in Computer Science, 1999, 1717, pp. 94–108.CrossRefGoogle Scholar
  12. 12.
    A. Avizienis. Signed-digit number representations r fast parallel arithmetic. IRE Transaction lectrononic Computers,EC(10):389–400, September 1961.Google Scholar
  13. 13.
    E. ÖztÜrk, E. Savaş, and B. Sunar,A Versatile ontgomery Multiplier Architecture with Characteristic Three upport.Under review, 2008.Google Scholar
  14. 14.
    E. Savaç, A. F. Tenca, M. E. Ciftcibasi, and Ç.K. Koç,“Multiplier architectures for \(GF(p)\) and \(GF(2^k)\)”, IEE Proceedings: Computers and Digital Techniques, 151(2): 147–160,March 2004.Google Scholar
  15. 15.
    S. E. Eldridge. A faster modular multiplication lgorithm. International Journal of Computational athematics, 40:63–68, 1991.MATHCrossRefGoogle Scholar
  16. 16.
    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
  17. 17.
    J.-H. Oh and S.-J. Moon. Modular multiplication ethod. IEE Proceedings, 145(4):317–318, July 1998.Google Scholar
  18. 18.
    C. D. Walter. Montgomery xponentitation needs no final subtractions. lectronic etters, 35(21):1831–1832, October 1999.Google Scholar
  19. 19.
    G. Hachez and J.-J. Quisquater. montgomery exponentiation with no final subtractions: Improved esults. In Cryptographic Hardware and Embedded Systems, Lecture Notes in Computer Science, No. 1965, pp.293–01. pringer-Verlag, Berlin, 2000.Google Scholar
  20. 20.
    B. S. Kaliski Jr., The Montgomery inverse and ts applications.IEEE Transactions on Computers, 44(8):1064–065, August 1995.MATHCrossRefGoogle Scholar
  21. 21.
    A. A.-A. Gutub, A. F. Tenca, E.Savaş, and Ç. K.Koç. Scalable and unified ardware to compute montgomeryinverse in \({GF}(p)\) and \(GF(2^n)\). In .B. S. Kaliski Jr., Ç.K. Koç and C. Paar, editors, Cryptographic Hardware and Embedded Systems, LNCS, pp.485–00, Springer-Verlag Berlin, 2002.Google Scholar
  22. 22.
    E. Savaş and Ç. K. Koç, Architecture for unifiedfield inversion with pplications in elliptic curve cryptography. InProc. vol. 3, he th IEEE International Conference onElectronics, Circuits and ystems ICECS 2002, pp. 1155–1158,Dubrovnik, Croatia, September 2002.Google Scholar
  23. 23.
    E. Savaç, M. Naseer, A. A-A. Gutub, and Ç. K. Koç.“Efficient Unified ontgomery Inversion with Multibit Shifting”,IEE roceedings: Computers and Digital Techniques, 152(4): 489–498, July 2005.CrossRefGoogle Scholar
  24. 24.
    E. Savaş and Ç. K. Koç, The Montgomery modularinverse - revisited. IEEE Transactions n Computers,49(7):763–766, July. 2000.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Sabanc¹ UniversitySabanc¹
  2. 2.City University of Istanbul & University of California Santa BarbaraSanta Barbara

Personalised recommendations