Exploring the Design Space of Prime Field vs. Binary Field ECC-Hardware Implementations

  • Erich Wenger
  • Michael Hutter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7161)

Abstract

In this paper, we answer the question whether binary extension field or prime-field based processors doing multi-precision arithmetic are better in the terms of area, speed, power, and energy. This is done by implementing and optimizing two distinct custom-made 16-bit processor designs and comparing our solutions on different abstraction levels: finite-field arithmetic, elliptic-curve operations, and on protocol level by implementing the Elliptic Curve Digital Signature Algorithm (ECDSA). On the one hand, our \(\mathbb{F}_{2^{m}}\) based processor outperforms the \(\mathbb{F}_p\) based processor by 19.7% in area, 69.6% in runtime, 15.9% in power, and 74.4% in energy when performing a point multiplication. On the other hand, our \(\mathbb{F}_p\) based processor (11.6kGE, 41.4,μW, 1,313kCycles, and 54.3μJ) improves the state-of-the-art in \(\mathbb{F}_{p_{192}}\) ECC hardware implementations regarding area, power, and energy results. After extending the designs for ECDSA (signature generation and verification), the area and power-consumption advantages of the \(\mathbb{F}_{2^{m}}\) based processor vanish, but it still is 1.5-2.8 times better in terms of energy and runtime.

Keywords

Hardware Implementation Elliptic Curve Cryptography ECC ECDSA Binary-Extension Field Prime Field 

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References

  1. 1.
    American National Standards Institute (ANSI). American National Standard X9.62-2005. Public Key Cryptography for the Financial Services Industry, The Elliptic Curve Digital Signature Algorithm (ECDSA) (2005)Google Scholar
  2. 2.
    Auer, A.: Scaling Hardware for Electronic Signatures to a Minimum. Master thesis, University of Technology Graz (October 2008)Google Scholar
  3. 3.
    Avanzi, R.M., Cohen, H., Doche, C., Frey, G., Lange, T., Nguyen, K., Vercauteren, F.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC (2005)Google Scholar
  4. 4.
    Batina, L., Mentens, N., Sakiyama, K., Preneel, B., Verbauwhede, I.: Low-Cost Elliptic Curve Cryptography for Wireless Sensor Networks. In: Buttyán, L., Gligor, V.D., Westhoff, D. (eds.) ESAS 2006. LNCS, vol. 4357, pp. 6–17. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Blake, I.F., Seroussi, G., Smart, N.P.: Elliptic Curves in Cryptography. London Mathematical Society Lecture Notes Series, vol. 265. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  6. 6.
    Bock, H., Braun, M., Dichtl, M., Hess, E., Heyszl, J., Kargl, W., Koroschetz, H., Meyer, B., Seuschek, H.: A Milestone Towards RFID Products Offering Asymmetric Authentication Based on Elliptic Curve Cryptography. Invited talk at RFIDsec 2008 (July 2008)Google Scholar
  7. 7.
    Cadence Design Systems, Inc., San Jose, California, United States (2011). The Cadence Design Systems Website, http://www.cadence.com/
  8. 8.
    de Rooij, P.: Efficient Exponentiation Using Precomputation and Vector Addition Chains. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 389–399. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  9. 9.
    El Gamal, T.: A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms. In: Blakely, G.R., Chaum, D. (eds.) CRYPTO 1984. LNCS, vol. 196, pp. 10–18. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  10. 10.
    Faraday Technology Corporation. Faraday FSA0A_C 0.13,μm ASIC Standard Cell Library (2004), http://www.faraday-tech.com
  11. 11.
    Fürbass, F., Wolkerstorfer, J.: ECC Processor with Low Die Size for RFID Applications. In: Proceedings of 2007 IEEE International Symposium on Circuits and Systems. IEEE (May 2007)Google Scholar
  12. 12.
    Großschädl, J., Savaş, E.: Instruction Set Extensions for Fast Arithmetic in Finite Fields GF(p) and GF(2m). In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 133–147. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Hankerson, D., Menezes, A.J., Vanstone, S.: Guide to Elliptic Curve Cryptography. Springer, Heidelberg (2004)MATHGoogle Scholar
  14. 14.
    Hein, D.: Elliptic Curve Cryptography ASIC for Radio Frequency Authentication. Master thesis, Technical University of Graz (April 2008)Google Scholar
  15. 15.
    Hein, D., Wolkerstorfer, J., Felber, N.: ECC Is Ready for RFID – A Proof in Silicon. In: Avanzi, R.M., Keliher, L., Sica, F. (eds.) SAC 2008. LNCS, vol. 5381, pp. 401–413. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Hutter, M., Feldhofer, M., Plos, T.: An ECDSA Processor for RFID Authentication. In: Ors Yalcin, S.B. (ed.) RFIDSec 2010. LNCS, vol. 6370, pp. 189–202. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Hutter, M., Joye, M., Sierra, Y.: Memory-Constrained Implementations of Elliptic Curve Cryptography in Co-Z Coordinate Representation. In: Nitaj, A., Pointcheval, D. (eds.) AFRICACRYPT 2011. LNCS, vol. 6737, pp. 170–187. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Itoh, T., Tsujii, S.: Effective recursive algorithm for computing multiplicative inverses in GF(2m). Electronic Letters 24(6), 334–335 (1988)MATHCrossRefGoogle Scholar
  19. 19.
    Joye, M., Yen, S.-M.: The Montgomery Powering Ladder. In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 291–302. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  20. 20.
    Kaliski, B.: The Montgomery Inverse and its Applications. IEEE Transactions on Computers 44(8), 1064–1065 (1995)MATHCrossRefGoogle Scholar
  21. 21.
    Kern, T., Feldhofer, M.: Low-Resource ECDSA Implementation for Passive RFID Tags. In: Proceedings of 17th IEEE International Conference on Electronics, Circuits and Systems (ICECS 2010), Athens, Greece, December 12-15, pp. 1236–1239. IEEE (2010)Google Scholar
  22. 22.
    Koblitz, N.: Elliptic Curve Cryptosystems. Mathematics of Computation 48, 203–209 (1987)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Koblitz, N.: A Course in Number Theory and Cryptography. Springer, Heidelberg (1994) ISBN 0-387-94293-9MATHCrossRefGoogle Scholar
  24. 24.
    Kumar, S.S., Paar, C.: Are standards compliant Elliptic Curve Cryptosystems feasible on RFID? In: Workshop on RFID Security (RFIDSec 2006), Graz, Austria, July 12-14 (2006)Google Scholar
  25. 25.
    Lee, Y.K., Sakiyama, K., Batina, L., Verbauwhede, I.: Elliptic-Curve-Based Security Processor for RFID. IEEE Transactions on Computers 57(11), 1514–1527 (2008)MathSciNetCrossRefGoogle Scholar
  26. 26.
    López, J., Dahab, R.: Improved Algorithms for Elliptic Curve Arithmetic in GF(2n). In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, pp. 201–212. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  27. 27.
    López, J., Dahab, R.: Fast Multiplication on Elliptic Curves over GF(2m). In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 316–327. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  28. 28.
    Miller, V.S.: Use of Elliptic Curves in Cryptography. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 417–426. Springer, Heidelberg (1986)Google Scholar
  29. 29.
    National Institute of Standards and Technology (NIST). FIPS-180-3: Secure Hash Standard (October 2008), http://www.itl.nist.gov/fipspubs/
  30. 30.
    National Institute of Standards and Technology (NIST). FIPS-186-3: Digital Signature Standard (DSS) (2009), http://www.itl.nist.gov/fipspubs/
  31. 31.
    Öztürk, E., Sunar, B., Savaş, E.: Low-Power Elliptic Curve Cryptography Using Scaled Modular Arithmetic. In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 92–106. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  32. 32.
    Satoh, A., Takano, K.: A Scalable Dual-Field Elliptic Curve Cryptographic Processor. IEEE Transactions on Computers 52(4), 449–460 (2003)CrossRefGoogle Scholar
  33. 33.
    Wenger, E., Feldhofer, M., Felber, N.: A 16-Bit Microprocessor Chip for Cryptographic Operations on Low-Resource Devices. In: Proceedings of Austrochip 2010, Villach, Austria, October 6, pp. 55–60 (2010) ISBN 978-3-200-01945-4Google Scholar
  34. 34.
    Wenger, E., Feldhofer, M., Felber, N.: Low-Resource Hardware Design of an Elliptic Curve Processor for Contactless Devices. In: Chung, Y., Yung, M. (eds.) WISA 2010. LNCS, vol. 6513, pp. 92–106. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  35. 35.
    Wenger, E., Hutter, M.: A Hardware Processor Supporting Elliptic Curve Cryptography for Less Than 9kGEs. In: Proceedings of the Tenth Smart Card Research and Advanced Application Conference, CARDIS 2011, Leuven, Belgium, September 15-16 (2011)Google Scholar
  36. 36.
    Wolkerstorfer, J.: Is Elliptic-Curve Cryptography Suitable for Small Devices? In: Workshop on RFID and Lightweight Crypto, Graz, Austria, July 13-15, pp. 78–91 (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Erich Wenger
    • 1
  • Michael Hutter
    • 1
  1. 1.Institute for Applied Information Processing and Communications (IAIK)Graz University of TechnologyGrazAustria

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