Mobile Networks and Applications

, Volume 12, Issue 4, pp 259–270 | Cite as

A State-of-the-art Elliptic Curve Cryptographic Processor Operating in the Frequency Domain

  • Selçuk Baktır
  • Sandeep Kumar
  • Christof Paar
  • Berk Sunar
Article

Abstract

We propose a novel area/time efficient elliptic curve cryptography (ECC) processor architecture which performs all finite field arithmetic operations in the discrete Fourier domain. The proposed architecture utilizes a class of optimal extension fields (OEF) GF(q m ) where the field characteristic is a Mersenne prime q = 2 n  − 1 and m = n. The main advantage of our architecture is that it achieves extension field modular multiplication in the discrete Fourier domain with only a linear number of base field GF(q) multiplications in addition to a quadratic number of simpler operations such as addition and bitwise rotation. We achieve an area between 25k and 50k equivalent gates for the implementations over OEFs of size 169, 289 and 361 bits. With its low area and high speed, the proposed architecture is well suited for ECC in small device environments such as sensor networks. The work at hand presents the first hardware implementation of a frequency domain multiplier suitable for ECC and the first hardware implementation of ECC in the frequency domain.

Keywords

elliptic curve cryptography (ECC) finite fields modular multiplication discrete Fourier domain 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bailey DV, Paar C (1998) Optimal extension fields for fast arithmetic in public-key algorithms. In: Krawczyk H (ed) Advances in cryptology—CRYPTO ’98, vol LNCS 1462. Springer, Berlin Heidelberg New York, pp 472–485Google Scholar
  2. 2.
    Bailey DV, Paar C (2001) Efficient arithmetic in finite field extensions with application in elliptic curve cryptography. J Cryptol 14(3):153–176MATHMathSciNetGoogle Scholar
  3. 3.
    Baktır S, Sunar B (2005) Finite field polynomial multiplication in the frequency domain with application to elliptic curve cryptography. Technical report. Worcester Polytechnic Institute, WorcesterGoogle Scholar
  4. 4.
    Baktır S, Sunar B (2006) Finite field polynomial multiplication in the frequency domain with application to elliptic curve cryptography. In: Proceedings of the 21st international symposium on computer and information sciences (ISCIS 2006). Lecture notes in computer science (LNCS), vol 4263. Springer, Berlin Heidelberg New York, pp 991–1001Google Scholar
  5. 5.
    Batina L, Örs SB, Preneel B, Vandewalle J (2003) Hardware architectures for public key cryptography. Integr VLSI J 34(6):1–64CrossRefGoogle Scholar
  6. 6.
    Blake IF, Seroussi G, Smart N (1999) Elliptic curves in cryptography. Mathematical Society Lecture Notes Series 265. Cambridge University Press, LondonMATHGoogle Scholar
  7. 7.
    Burrus CS, Parks TW (1985) DFT/FFT and convolution algorithms. Wiley, New YorkGoogle Scholar
  8. 8.
    Kalach K, David JP (2005) Hardware implementation of large number multiplication by FFT with modular arithmetic. In: Proceedings of the 3rd international IEEE-NEWCAS conference. IEEE, Piscataway, pp 267–270Google Scholar
  9. 9.
    Koblitz N (1987) Elliptic curve cryptosystems. Math Comput 48:203–209MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kumar S, Girimondo M, Weimerskirch A, Paar C, Patel A, Wander AS (2003) Embedded end-to-end wireless security with ECDH key exchange. In: 46th IEEE midwest symposium on circuits and systems, Cairo, pp 27–30 December 2003Google Scholar
  11. 11.
    Lee M-K, Kim KT, Kim H, Kim DK (2006) Efficient hardware implementation of elliptic curve cryptography over GF(p m). In: Proceedings of the 6th international workshop on information security applications (WISA 2005). Lecture notes in computer science (LNCS), vol 3786. Springer, Berlin Heidelberg New York, pp 207–217Google Scholar
  12. 12.
    Menezes AJ, van Oorschot PC, Vanstone SA (1997) Handbook of applied cryptography. CRC, Boca RatonMATHGoogle Scholar
  13. 13.
    Miller V (1986) Uses of elliptic curves in cryptography. In: Williams HC (ed) Advances in cryptology, CRYPTO ’85, vol LNCS 218. Springer, Berlin Heidelberg New York, pp 417–426Google Scholar
  14. 14.
    Öztürk E, Sunar B, Savas E (2004) Low-power elliptic curve cryptography using scaled modular arithmetic. In: Proceedings of the workshop on cryptographic hardware and embedded systems (CHES 2004). Lecture notes in computer science (LNCS), vol 3156. Springer, Berlin Heidelberg New York, pp 92–106Google Scholar
  15. 15.
    Pollard JM (1971) The fast fourier transform in a finite field. Math Comput 25:365–374MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Rader CM (1972) Discrete convolutions via mersenne transforms. IEEE Trans Comput C-21(12):1269–1273MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Rivest RL, Shamir A, Adleman L (1978) A method for obtaining digital signatures and public-key cryptosystems. Commun ACM 21(2):120–126MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Satoh A, Takano K (2003) A scalable dual-field elliptic curve cryptographic processor. IEEE Trans Comput 52(4):1–64CrossRefGoogle Scholar
  19. 19.
    Tolimieri R, An M, Lu C (1989) Algorithms for discrete fourier transform and convolution. Springer, Berlin Heidelberg New YorkMATHGoogle Scholar
  20. 20.
    Woodbury A, Bailey DV, Paar C (2000) Elliptic curve cryptography on smart cards without coprocessors. In: IFIP CARDIS 2000, fourth smart card research and advanced application conference. Kluwer, Bristol, pp 20–22Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Selçuk Baktır
    • 1
  • Sandeep Kumar
    • 2
  • Christof Paar
    • 2
  • Berk Sunar
    • 1
  1. 1.Cryptography & Information Security LaboratoryWPIWorcesterUSA
  2. 2.Communication Security Group (COSY)Ruhr-University BochumBochumGermany

Personalised recommendations