Subscribe

Issue No.04 - April (2013 vol.62)

pp: 744-757

R. Azarderakhsh , Dept. of Electr. & Comput. Eng., Univ. of Western Ontario, London, ON, Canada

A. Reyhani-Masoleh , Dept. of Electr. & Comput. Eng., Univ. of Western Ontario, London, ON, Canada

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TC.2012.22

ABSTRACT

The extensive rise in the number of resource constrained wireless devices and the needs for secure communications with the servers imply fast and efficient cryptographic computations for both parties. Efficient hardware implementation of arithmetic operations over finite field using Gaussian normal basis is attractive for public key cryptography as it provides free squarings. In this paper, we first present two low-complexity digit-level multiplier architectures. It is shown that the proposed multipliers outperform the existing Gaussian normal basis (GNB) multiplier structures available in the literature. Then, for the first time, using these two architectures, we propose a new digit-level hybrid multiplier which performs two successive multiplications with the same latency as the one for one multiplication. We have studied the efficiency of the proposed hybrid architecture in terms of area and time delay for different digit sizes. The main advantage of this new hybrid architecture is to speed up exponentiation and point multiplication whenever double-multiplication is required and the traditional schemes fail due to the data dependencies. We have investigated the applicability of the proposed hybrid structure to reduce the latency of exponentiation-based cryptosystems. Our analysis and timing results show that the expected acceleration in double-exponentiation is considerable. Prototypes of the presented low-complexity multiplier architectures and the proposed hybrid architecture are implemented and experimental results are presented.

INDEX TERMS

public key cryptography, Gaussian processes, data dependency, low-complexity digit-level multiplier architectures, single multiplications, hybrid-double multiplications, Gaussian normal basis, resource constrained wireless devices, communication security, cryptographic computations, public key cryptography, GNB multiplier structures, digit-level hybrid multiplier, exponentiation-based cryptosystems, point multiplication, Gaussian processes, Computer architecture, Registers, Logic gates, Complexity theory, Clocks, Cryptography, double-exponentiation, Cryptosystems, Gaussian normal basis, double-multiplication, digit-level multiplier

CITATION

R. Azarderakhsh, A. Reyhani-Masoleh, "Low-Complexity Multiplier Architectures for Single and Hybrid-Double Multiplications in Gaussian Normal Bases",

*IEEE Transactions on Computers*, vol.62, no. 4, pp. 744-757, April 2013, doi:10.1109/TC.2012.22REFERENCES

- [1] V.S. Miller, "Use of Elliptic Curves in Cryptography,"
Proc. Advances in Cryptology (Crypto), pp. 417-426, 1986.- [2] N. Koblitz, "Elliptic Curve Cryptosystems,"
Math. of Computation, vol. 48, pp. 203-209, 1987.- [3] T.E. Gamal, "A Public Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms,"
IEEE Trans. Information Theory, vol. 31, no. 4, pp. 469-472, July 1985.- [4] W. Diffie and M. Hellman, "New Directions in Cryptography,"
IEEE Trans. Information Theory, vol. 22, no. 6, pp. 644-654, Nov. 1976.- [5] D.W. Ash, I.F. Blake, and S.A. Vanstone, "Low Complexity Normal Bases,"
Discrete Applied Math., vol. 25, no. 3, pp. 191-210, 1989.- [6] IEEE Std 1363-2000, "IEEE Standard Specifications for Public-Key Cryptography," Jan. 2000.
- [7] US Dept. of Commerce/NIST, "National Institute of Standards and Technology,"
Digital Signature Standard, FIPS Publications 186-2, Jan. 2000.- [8] J. Massey and J. Omura,
Computational Method and Apparatus for Finite Arithmetic, US Patent 4587627, Washington, D.C., 1986.- [9] G. Feng, "A VLSI Architecture for Fast Inversion in ${GF}(2^m)$ ,"
IEEE Trans. Computers, vol. 38, no. 10, pp. 1383-1386, Oct. 1989.- [10] T. Beth and D. Gollman, "Algorithm Engineering For Public Key Algorithms,"
IEEE J. Selected Areas in Communications, vol. 7, no. 4, pp. 458-466, May 1989.- [11] C. Lee, P. Meher, and J. Patra, "Concurrent Error Detection in Bit-Serial Normal Basis Multiplication Over ${GF}(2^m)$ Using Multiple Parity Prediction Schemes,"
IEEE Trans. Very Large Scale Integration (VLSI) Systems, vol. 18, no. 8, pp. 1234-1238, Aug. 2010.- [12] W. Geiselmann and D. Gollmann, "Symmetry and Duality in Normal Nasis Multiplication,"
Proc. Sixth Symp. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC), pp. 230-238, July 1989.- [13] G.B. Agnew, R.C. Mullin, I.M. Onyszchuk, and S.A. Vanstone, "An Implementation for a Fast Public-Key Cryptosystem,"
J. Cryptology, vol. 3, no. 2, pp. 63-79, 1991.- [14] A. Reyhani-Masoleh and M.A. Hasan, "Efficient Digit-serial Normal Basis Multipliers over Binary Extension Fields,"
ACM Trans. Embedded Computing Systems, vol. 3, no. 3, pp. 575-592, Aug. 2004.- [15] S. Kwon, K. Gaj, C.H. Kim, and C.P. Hong, "Efficient Linear Array for Multiplication in ${GF}(2^m)$ Using a Normal Basis for Elliptic Curve Cryptography,"
Proc. Workshop Cryptographic Hardware and Embedded Systems (CHES), pp. 76-91, Aug. 2004.- [16] A. Reyhani-Masoleh, "Efficient Algorithms and Architectures for Field Multiplication Using Gaussian Normal Bases,"
IEEE Trans. Computers, vol. 55, no. 1, pp. 34-47, Jan. 2006.- [17] A.H. Namin, H. Wu, and M. Ahmadi, "A Word-Level Finite Field Multiplier Using Normal Basis,"
IEEE Trans. Computers, vol. 60, no. 6, pp. 890-895, June 2010.- [18] C. Lee and P. Chang, "Digit-Serial Gaussian Normal Basis Multiplier over ${GF}(2^m)$ Using Toeplitz Matrix-Approach,"
Proc. Int'l Conf. Computational Intelligence and Software Eng. (CiSE), pp. 1-4, 2009.- [19] Ç. K. Koç and B. Sunar, "An Efficient Optimal Normal Basis Type II Multiplier over ${GF}(2^m)$ ,"
IEEE Trans. Computers, vol. 50, no. 1, pp. 83-87, Jan. 2001.- [20] M. Hasan, M. Wang, and V. Bhargava, "A modified Massey-Omura Parallel Multiplier for a Class of Finite Fields,"
IEEE Trans. Computers, vol. 42, no. 10, pp. 1278-1280, Oct. 1993.- [21] A. Reyhani-Masoleh and M.A. Hasan, "A New Construction of Massey-Omura Parallel Multiplier over ${GF}(2^m)$ ,"
IEEE Trans. Computers, vol. 51, no. 5, pp. 511-520, May 2002.- [22] L. Gao and G.E. Sobelman, "Improved VLSI Designs for Multiplication and Inversion in ${GF}(2^m)$ over Normal Bases,"
Proc. IEEE 13th Ann. Int'l ASIC/SOC Conf., pp. 97-101, 2000.- [23] K. Järvinen and J. Skyttä, "On Parallelization of High-Speed Processors for Elliptic Curve Cryptography,"
IEEE Trans. Very Large Scale Integration (VLSI) Systems, vol. 16, no. 9, pp. 1162-1175, Sept. 2008.- [24] C.H. Kim, S. Kwon, and C.P. Hong, "FPGA Implementation of High Performance Elliptic Curve Cryptographic Processor over ${GF}(2^{163})$ ,"
J. System Architecture, vol. 54, no. 10, pp. 893-900, 2008.- [25] R. Azarderakhsh and A. Reyhani-Masoleh, "A Modified Low Complexity Digit-Level Gaussian Normal Basis Multiplier,"
Proc. Third Int'l Workshop Arithmetic of Finite Fields (WAIFI), pp. 25-40, June 2010.- [26] C.-P. Schnorr, "Efficient Signature Generation by Smart Cards,"
J. Cryptology, vol. 4, no. 3, pp. 161-174, 1991.- [27] A. Menezes, I. Blake, S. Gao, R. Mullin, S. Vanstone, and T. yaghoobian,
Applications of Finite Fields. Kluwer Academic Publisher, 1993.- [28] C.-Y. Lee, "Concurrent Error Detection Architectures for Gaussian Normal Basis Multiplication over ${GF}(2^m)$ ,"
Integration, the VLSI J., vol. 43, no. 1, pp. 113-123, 2010.- [29] M. Elia and M. Leone, "On the Inherent Space Complexity of Fast Parallel Multipliers for ${GF}(2^m)$ ,"
IEEE Trans. Computers, vol. 51, no. 3, pp. 346-351, Mar. 2002.- [30] C. Wang and D. Pei, "A VLSI Design for Computing Exponentiations in ${GF}(2^m)$ and Its Application to Generate Pseudorandom Number Sequences,"
IEEE Trans. Computers, vol. 39, no. 2, pp. 258-262, Feb. 1990.- [31] C. Lee, J. Lin, and C. Chiou, "Scalable and Systolic Architecture for Computing Double Exponentiation over ${GF}(2^m)$ ,"
Acta Applicandae Mathematicae, vol. 93, no. 1, pp. 161-178, 2006.- [32] J.H. Cheon, S. Jarecki, T. Kwon, and M.-K. Lee, "Fast Exponentiation Using Split Exponents,"
IEEE Trans. Information Theory, vol. 57, no. 3, pp. 1816-1826, Mar. 2011.- [33] J. Fan, D. Bailey, L. Batina, T. Guneysu, C. Paar, and I. Verbauwhede, "Breaking Elliptic Curves Cryptosystems using Reconfigurable Hardware,"
Proc. 20th Int'l Conf. Field Programmable Logic and Applications (FPL), pp. 133-138, 2010.- [34] Certicom, "Certicom ECC Chalenge," www.certicom.com, 1997.
- [35] T. Itoh and S. Tsujii, "A Fast Algorithm for Computing Multiplicative Inverses in ${GF}(2^m)$ Using Normal Bases,"
Information Computing, vol. 78, no. 3, pp. 171-177, 1988. |