DOI QR코드

DOI QR Code

Low-latency Montgomery AB2 Multiplier Using Redundant Representation Over GF(2m))

GF(2m) 상의 여분 표현을 이용한 낮은 지연시간의 몽고메리 AB2 곱셈기

  • Received : 2016.11.14
  • Accepted : 2016.12.27
  • Published : 2017.02.28

Abstract

Finite field arithmetic has been extensively used in error correcting codes and cryptography. Low-complexity and high-speed designs for finite field arithmetic are needed to meet the demands of wider bandwidth, better security and higher portability for personal communication device. In particular, cryptosystems in GF($2^m$) usually require computing exponentiation, division, and multiplicative inverse, which are very costly operations. These operations can be performed by computing modular AB multiplications or modular $AB^2$ multiplications. To compute these time-consuming operations, using $AB^2$ multiplications is more efficient than AB multiplications. Thus, there are needs for an efficient $AB^2$ multiplier architecture. In this paper, we propose a low latency Montgomery $AB^2$ multiplier using redundant representation over GF($2^m$). The proposed $AB^2$ multiplier has less space and time complexities compared to related multipliers. As compared to the corresponding existing structures, the proposed $AB^2$ multiplier saves at least 18% area, 50% time, and 59% area-time (AT) complexity. Accordingly, it is well suited for VLSI implementation and can be easily applied as a basic component for computing complex operations over finite field, such as exponentiation, division, and multiplicative inverse.

Acknowledgement

Supported by : 한국연구재단

References

  1. A.J. Menezes, P.C. van Oorschot, S.A. Vanstone, Handbook of Applied Cryptography, Boca Raton, FL, CRC Press, 1996.
  2. R.E. Blahut, Theory and Practice of Error Control Codes, Reading, MA, Addison-Wesley, 1983.
  3. N. Kobliz, "Elliptic curve cryptography," Math. Computation, Vol. 48, No. 177, pp. 203-209, 1987. https://doi.org/10.1090/S0025-5718-1987-0866109-5
  4. P. Montgomery, "Modular multiplication without trial division," Mathematics of Computation, Vol. 44, No. 170, pp. 519-521, 1985. https://doi.org/10.1090/S0025-5718-1985-0777282-X
  5. C.K. Koc, T. Acar, "Montgomery multiplication in GF($2^k$)," Designs Codes and Cryptography, vol. 14, pp. 57-69, 1998. https://doi.org/10.1023/A:1008208521515
  6. C.Y. Lee, J.S. Horng, I.C. Jou, "Low-complexity bit-parallel systolic Montgomery multipliers for special classes of GF($2^m$)," IEEE Transactions on Computers, Vol. 54, No. 9, pp. 1061-1070, 2005. https://doi.org/10.1109/TC.2005.147
  7. C.W. Chiou, C.Y. Lee, A.W. Deng, J.M. Lin, "Concurrent error detection in Montgomery multiplication over GF($2^m$)," IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, Vol. E89-A, No. 2, pp. 566-574, 2006. https://doi.org/10.1093/ietfec/e89-a.2.566
  8. A. Hariri, A. Reyhani-Masoleh, "Bit-serial and bit-parallel Montgomery multiplication and squaring over GF($2^m$)," IEEE Transactions on Computers, Vol. 58, No. 10, pp. 1332-45, 2009. https://doi.org/10.1109/TC.2009.70
  9. A. Hariri, A. Reyhani-Masoleh, "Concurrent error detection in Montgomery multiplication over binary extension fields," IEEE Transactions on Computers, Vol. 60, No. 9, pp. 1341-53, 2011. https://doi.org/10.1109/TC.2010.258
  10. K.W. Kim, W.J. Lee, "Efficient cellular automata based Montgomery $AB^2$ multipliers over GF($2^m$)," IETE Technical Review, Vol. 31, No. 1, pp. 92-102, 2014. https://doi.org/10.1080/02564602.2014.891383
  11. K.W. Kim, J.C. Jeon, "Polynomial basis multiplier using cellular systolic architecture," IETE Journal of Research, Vol. 60, No. 2, pp. 194-199, 2014. https://doi.org/10.1080/03772063.2014.914699
  12. S.H. Choi, K.J. Lee, "Low complexity semi-systolic multiplication architecture over GF($2^m$)," IEICE Electron. Express, Vol. 11, No. 20, pp. 20140713, 2014. https://doi.org/10.1587/elex.11.20140713
  13. K.W. Kim, J.C. Jeon, "A semi-systolic Montgomery multiplier over GF($2^m$)," IEICE Electonics Express, Vol. 12, No. 21, pp. 20150769, 2015. https://doi.org/10.1587/elex.12.20150769
  14. H.H. Lee, K.W. Kim, "Efficient semi-systolic finite field multiplier using redundant basis," International Journal of Computer, Electrical, Automation, Control and Information Engineering, Vol. 10, No. 10, pp. 1614-1618, 2016.
  15. S.W. Wei, "A systolic power-sum circuit for GF($2^m$)," IEEE Transactions on Computers, Vol. 43, No. 2, pp. 226-229, 1994. https://doi.org/10.1109/12.262128
  16. C.L. Wang, J.H. Guo, "New systolic arrays for $C+AB^2$, inversion, and division in GF($2^m$)," IEEE Transactions on Computers, Vol. 49, No. 10, pp. 1120-1125, 2000. https://doi.org/10.1109/12.888047
  17. C.H. Liu, N.F. Huang, C.Y. Lee, "Computation of $AB^2$ multiplier in GF($2^m$) using an efficient low-complexity cellular architecture," IEICE Transactions on Fundamentals of Electronics, Vol. E83-A, No. 12, pp. 2657-2663, 2000.
  18. C.Y. Lee, E.H. Lu, L.F. Sun, "Low-complexity bit-parallel systolic architecture for computing $AB^2+C$ in a class of finite field GF($2^m$)," IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, Vol. 48, No. 5, pp. 519-523, 2001. https://doi.org/10.1109/82.938363
  19. Y.R. Ting, E.H. Lu, J.Y. Lee, "Low complexity bit-parallel systolic architecture for computing $C+AB^2$ over a class of GF(2m)," INTEGRATION, the VLSI journal, Vol. 37, No. 3, pp. 167-176, 2004. https://doi.org/10.1016/j.vlsi.2004.01.003
  20. C.Y. Lee, A.W. Chiou, J.M. Lin, "Low-complexity bit-parallel systolic architectures for computing $A(x)B^2(x)$ over GF($2_m$)," IEEE Proceedings of Circuits Devices and Systtems, Vol. 153, No. 4, pp. 399-406, 2006. https://doi.org/10.1049/ip-cds:20050188
  21. K.W. Kim, W.J. Lee, "Low-complexity parallel and serial systolic architectures for $AB^2$ multiplication in GF($2_m$)," IETE Technical Review, Vol. 30, No. 2, pp. 134-141, 2013. https://doi.org/10.4103/0256-4602.110552
  22. K.W. Kim, W.J. Lee, "An efficient parallel systolic array for $AB^2$ over GF($2_m$)," IEICE Electronics Express, Vol. 10, No. 20, pp. 20130585, 2013. https://doi.org/10.1587/elex.10.20130585
  23. K.W. Kim, W.J. Lee, "Efficient cellular automata based Montgomery $AB^2$ multipliers over GF($2_m$)," IETE Technical Review, Vol 31, No. 1, pp. 92-102, 2014. https://doi.org/10.1080/02564602.2014.891383
  24. G. Drolet, "A new representation of elements of finite fields yielding small complexity arithmetic circuits," IEEE Transactions on Computers, Vol. 47, No. 9, pp. 938-946, 1998. https://doi.org/10.1109/12.713313
  25. H. Wu, M.A. Hasan, I.F. Blake, S. Gao, "Finite field multiplier using redundant representation," IEEE Transactions on Computers, Vol. 51, No. 11, pp. 1306-1316, 2002. https://doi.org/10.1109/TC.2002.1047755