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A Modified Low Complexity Digit-Level Gaussian Normal Basis Multiplier

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Arithmetic of Finite Fields (WAIFI 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6087))

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Abstract

Gaussian normal bases have been included in a number of standards, such as [1] and NIST [2] for elliptic curve digital signature algorithm (ECDSA). Among different finite field operations used in this algorithm, multiplication is the main operation. In this paper, we consider type T Gaussian normal basis (GNB) multipliers over GF(2m), where m is odd. Such fields include five binary fields recommended by NIST for ECDSA. A modified digit-level GNB multiplier over GF(2m) is proposed in this paper. For T > 2, a complexity reduction algorithm is proposed to reduce the number of XOR gates without increasing the gate delay of the digit-level multiplier. The original and modified digit-level GNB multipliers are implemented on the Xilinx® Virtex5TM FPGA family for different digit sizes. It is shown that the modified digit-level GNB multiplier requires lower space complexity with almost the same delay as compared to the original type T, T > 2, GNB multiplier. Moreover, the bit-parallel GNB multiplier obtained from the proposed modified digit-level multiplier has the least space and time complexities among the existing fast bit-parallel type T GNB multipliers for T > 2.

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References

  1. IEEE Std 1363-2000: IEEE Standard Specifications for Public-Key Cryptography (January 2000)

    Google Scholar 

  2. U.S. Department of Commerce/NIST: Digital Signature Standards (DSS). Federal Information Processing Standards Publications (2000)

    Google Scholar 

  3. 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 

  4. Koblitz, N.: Elliptic curve cryptosystems. Mathematics of Computation 48, 203–209 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dimitrov, V.S., Järvinen, K.U., Jacobson Jr., M.J., Chan, W.F., Huang, Z.: Provably Sublinear Point Multiplication on Koblitz Curves and its Hardware Implementation. IEEE Transaction on Computers 57(11), 1469–1481 (2008)

    Article  Google Scholar 

  6. Järvinen, K., Skyttä, J.: On Parallelization of High-Speed Processors for Elliptic Curve Cryptography. IEEE Transactions on Very Large Scale Integration (VLSI) Systems 16(9), 1162–1175 (2008)

    Article  Google Scholar 

  7. Kim, C.H., Kwon, S., Hong, C.P.: FPGA Implementation of High Performance Elliptic Curve Cryptographic Processor over GF(2163). Journal of System Architcture 54(10), 893–900 (2008)

    Article  Google Scholar 

  8. Massey, J., Omura, J.: Computational Method and Apparatus for Finite Arithmetic. US Patent (4587627) (1986)

    Google Scholar 

  9. Agnew, G.B., Mullin, R.C., Onyszchuk, I.M., Vanstone, S.A.: An Implementation for a Fast Public-Key Cryptosystem. Journal of Cryptology 3(2), 63–79 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kwon, S., Gaj, K., Kim, C.H., Hong, C.P.: Efficient Linear Array for Multiplication in GF(2m) using a Normal Basis for Elliptic Curve Cryptography. In: Joye, M., Quisquater, J.-J. (eds.) CHES 2004. LNCS, vol. 3156, pp. 76–91. Springer, Heidelberg (2004)

    Google Scholar 

  11. Wang, C.C., Truong, T.K., Shao, H.M., Deutsch, L.J., Omura, J.K., Reed, I.S.: VLSI Architectures for Computing Multiplications and Inverses in GF(2m). IEEE Transaction on Computers 34(8), 709–717 (1985)

    Article  MATH  Google Scholar 

  12. Gao, L., Sobelman, G.E.: Improved VLSI Designs for Multiplication and Inversion in GF(2M) over normal bases. In: Proceedings of 13th Annual IEEE International ASIC/SOC Conference, pp. 97–101 (2000)

    Google Scholar 

  13. Reyhani-Masoleh, A., Hasan, M.A.: A New Construction of Massey-Omura Parallel Multiplier over GF(2m). IEEE Transactions on Computers 51(5), 511–520 (2002)

    Article  MathSciNet  Google Scholar 

  14. Koç, Ç.K., Sunar, B.: An Efficient Optimal Normal Basis Type II Multiplier over GF(2m). IEEE Transaction on Computers 50(1), 83–87 (2001)

    Article  Google Scholar 

  15. Reyhani-Masoleh, A.: Efficient Algorithms and Architectures for Field Multiplication Using Gaussian Normal Bases. IEEE Transaction On Computers, 34–47 (2006)

    Google Scholar 

  16. Fan, H., Hasan, M.: Subquadratic computational complexity schemes for extended binary field multiplication using optimal normal bases. IEEE Transactions on Computers 56(10), 1435 (2007)

    Article  MathSciNet  Google Scholar 

  17. Gathen, J., Shokrollahi, A., Shokrollahi, J.: Efficient multiplication using type 2 optimal normal bases. In: Carlet, C., Sunar, B. (eds.) WAIFI 2007. LNCS, vol. 4547, pp. 55–68. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  18. Lidl, R., Niederreiter, H.: Introduction to Finite Fields and Their Applications. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  19. Mullin, R.C., Onyszchuk, I.M., Vanstone, S.A., Wilson, R.M.: Optimal Normal Bases in GF(p n). Discrete Appl. Math. 22(2), 149–161 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gao, S., Lenstra, H.W.: Optimal Normal Bases. Designs, Codes and Cryptography 2, 315–323 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ash, D.W., Blake, I.F., Vanstone, S.A.: Low Complexity Normal Bases. Discrete Applied Mathematics 25(3), 191–210 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gustafsson, O., Olofsson, M.: Complexity reduction of constant matrix computations over the binary field. In: Carlet, C., Sunar, B. (eds.) WAIFI 2007. LNCS, vol. 4547, pp. 103–115. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, The University of Western Ontario, London, ON, Canada, N6A 5B9

    Reza Azarderakhsh & Arash Reyhani-Masoleh

Authors
  1. Reza Azarderakhsh
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  2. Arash Reyhani-Masoleh
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Editor information

Editors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    M. Anwar Hasan

  2. Department of Informatics, HIB, University of Bergen, PB 7803, 5020, Bergen, Norway

    Tor Helleseth

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Azarderakhsh, R., Reyhani-Masoleh, A. (2010). A Modified Low Complexity Digit-Level Gaussian Normal Basis Multiplier. In: Hasan, M.A., Helleseth, T. (eds) Arithmetic of Finite Fields. WAIFI 2010. Lecture Notes in Computer Science, vol 6087. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13797-6_3

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  • DOI: https://doi.org/10.1007/978-3-642-13797-6_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13796-9

  • Online ISBN: 978-3-642-13797-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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