The Community for Technology Leaders
RSS Icon
Issue No.02 - February (2011 vol.60)
pp: 189-201
Vassil S. Dimitrov , University of Calgary, Calgary
Kimmo U. Järvinen , Aalto University, Aalto
Jithra Adikari , University of Calgary, Calgary
In this paper, we shall introduce several new algorithms for integer multiplication that are based on specific multiple-radix representation of one of the multiplicands. We provide extensive theoretical analysis and experimental results for multipliers based on the new representations on 0.18 {\rm \mu m} CMOS technology. They provide a clear picture about the advantages of the new method in 64-bit hardware implementations compared to array-based classical multiplier and radix-8-based multiplier. The proposed multipliers have better area and power consumption compared to reference multipliers.
Integer multiplication, multiple-radix representation, double-base number system.
Vassil S. Dimitrov, Kimmo U. Järvinen, Jithra Adikari, "Area-Efficient Multipliers Based on Multiple-Radix Representations", IEEE Transactions on Computers, vol.60, no. 2, pp. 189-201, February 2011, doi:10.1109/TC.2010.200
[1] A.N. Kolmogorov, "Asymptotic Characteristics of Some Completely Bounded Metric Spaces," Doklady Akademii Nauk SSSR, vol. 108, pp. 585-589, 1956.
[2] A. Karatsuba and Y. Ofman, "Multiplication of Multidigit Numbers on Automata," Soviet Physics Doklady, vol. 7, no. 7, pp. 595-596, Jan. 1963.
[3] D.E. Knuth, Art of Computer Programming, Vol. 2: Seminumerical Algorithms, third ed. Addison-Wesley Professional, Nov. 1997.
[4] D. Zuras, "More on Squaring and Multiplying Large Integers," IEEE Trans. Computers , vol. 43, no. 8, pp. 899-908, Aug. 1994.
[5] V. Strassen, "Gaussian Elimination Is Not Optimal," Numerische Mathematik, vol. 13, no. 3, pp. 354-356, 1969.
[6] A. Schönhage and, V. Strassen, "Schnelle Multiplikation Großer Zahlen," Computing, vol. 7, pp. 281-292, 1971.
[7] M. Fürer, "Faster Integer Multiplication," Proc. 39th Ann. ACM Symp. Theory of Computing, pp. 57-66, 2007.
[8] S.A. Cook, "On the Minimal Computation Time of Functions," PhD dissertation, Harvard Univ., 1966.
[9] V. Lefèvre, "Multiplication by an Integer Constant," 2000.
[10] R. Pinch, "Asymptotic Upper Bound for Multiplier Design," Electronics Letters, vol. 32, no. 5, pp. 420-421, Feb. 1996.
[11] V.S. Dimitrov, L. Imbert, and A. Zakaluzny, "Multiplication by a Constant Is Sublinear," Proc. 18th IEEE Symp. Computer Arithmetic (ARITH18), pp. 261-268, June 2007.
[12] V.S. Dimitrov, L. Imbert, and P.K. Mishra, "The Double-Base Number System and Its Application to Elliptic Curve Cryptography," Math. Computation, vol. 77, no. 262, pp. 1075-1104, Dec. 2007.
[13] P. Erdős, C. Pomerance, and E. Schmutz, "Carmichael's Lambda Function," Acta Arithmetica, vol. 58, no. 4, pp. 363-385, 1991.
[14] V.S. Dimitrov and E.W. Howe, "Lower Bounds on the Lengths of Double-Base Representations," Jan 2010.
[15] V.S. Dimitrov, G.A. Jullien, and W.C. Miller, "An Algorithm for Modular Exponentiation," Information Processing Letters, vol. 66, no. 3, pp. 155-159, 1998.
[16] R. Tijdeman, "On the Maximal Distance between Integers Composed of Small Primes," Compositio Mathematica, vol. 28, pp. 159-162, 1974.
[17] A. Baker, "Linear Forms in the Logarithms of Algebraic Numbers IV," Mathematika, vol. 16, pp. 204-216, 1968.
[18] M. Mignotte and M. Waldschmidt, "Linear Forms in Two Logarithms and Schneider's Method. III," Mathematische Annalen, vol. 231, pp. 241-267, 1990.
[19] V.S. Dimitrov and T.V. Cooklev, "Hybrid Algorithm for the Computation of the Matrix Polynomial ${I}+{A}+ \cdots + {A}^{N-1}$ ," IEEE Trans. Circuits and Systems I: Fundamental Theory and Application, vol. 42, no. 7, pp. 377-380, July 1995.
[20] V.S. Dimitrov and G.A. Jullien, "Loading the Bases: A New Number Representation with Applications," IEEE Circuits and Systems Magazine, vol. 3, no. 2, pp. 6-23, Nov. 2003.
[21] G. Jullien, V.S. Dimitrov, B. Li, W.C. Miller, A. Lee, and M. Ahmadi, "A Hybrid DBNS Processor for DSP Computation," Proc. IEEE Int'l Symp. Circuits and Systems (ISCAS), vol. 1, pp. 5-8, July 1999.
[22] R. Muscedere, V.S. Dimitrov, G.A. Jullien, and W.C. Miller, "Efficient Techniques for Binary-to-Multidigit Multidimensional Logarithmic Number System Conversion Using Range-Addressable Lookup Tables," IEEE Trans. Computers, vol. 54, no. 3, pp. 257-271, Mar. 2005.
[23] R. Muscedere, V.S. Dimitrov, G.A. Jullien, and W.C. Miller, "A Low-Power Two-Digit Multidimensional Logarithmic Number System Filterbank Architecture for a Digital Hearing Aid," EURASIP J. Applied Signal Processing, vol. 18, pp. 3015-3025, 2005.
[24] R. Muscedere, V.S. Dimitrov, G.A. Jullien, W.C. Miller, and M. Ahmadi, "On Efficient Techniques for Difficult Operations in One and Two-Digit DBNS Index Calculus," Proc. 34th Asilomar Conf. Signals, Systems and Computers, vol. 2, pp. 870-874, 2000.
[25] S.M. Kilambi and B. Nowrouzian, "A Novel Genetic Algorithm for Optimization of FRM Digital Filters over DBNS Multiplier Coefficient Space Based on Correlative Roulette Selection," Proc. IEEE Int'l Symp. Signal Processing and Information Technology, pp. 228-231, Aug. 2006.
[26] P. Mercier, S.M. Kilambi, and B. Nowrouzian, "Optimization of FRM FIR Digital Filters over CSD and CDBNS Multiplier Coefficient Spaces Employing a Novel Genetic Algorithm," J. Computers, vol. 2, no. 7, pp. 20-31, 2007.
[27] M. Azarmehr and R. Muscedere, "A Simple Central Processing Unit with Multi-Dimensional Logarithmic Number System Extensions," Proc. IEEE Int'l Conf. Application-Specific Systems, Architectures and Processors (ASAP), pp. 342-345, July 2007.
[28] V.S. Dimitrov, K.U. Järvinen, M.J. JacobsonJr., W.F. Chan, and Z. Huang, "Provably Sublinear Point Multiplication on Koblitz Curves and Its Hardware Implementation," IEEE Trans. Computers, vol. 57, no. 11, pp. 1469-1481, Nov. 2008.
[29] V.S. Dimitrov, L. Imbert, and P.K. Mishra, "Efficient and Secure Elliptic Curve Point Multiplication Using Double-Base Chains," Proc. Int'l Conf. Theory and Application of Cryptology and Information Security (ASIACRYPT), pp. 59-78, 2005.
[30] R. Avanzi, V.S. Dimitrov, C. Doche, and F. Sica, "Extending Scalar Multiplication Using Double Bases," Proc. Int'l Conf. Theory and Application of Cryptology and Information Security (ASIACRYPT), pp. 130-144, 2006.
[31] M. Ciet and F. Sica, "An Analysis of Double Base Number Systems and a Sublinear Scalar Multiplication Algorithm," Proc. Int'l Conf. Cryptology in Malayasia (MYCRPT), pp. 171-182, 2005.
[32] C. Doche and L. Imbert, "Extended Double-Base Number System with Applications to Elliptic Curve Cryptosystem," Proc. Int'l Conf. Cryptology in India (INDOCRYPT), pp. 335-348, 2006.
[33] V. Berthé and L. Imbert, "On Converting Number to the Double-Base Number System," Proc. SPIE Conf., pp. 70-78, 2004.
[34] V.S. Dimitrov, T.V. Cooklev, "Two Algorithms for Modular Exponentiation Using Nonstandard Arithmetic," IEICE Trans. Fundamentals Electronics, Comm. and Computer Sciences , vol. E78-A, pp. 82-87, 1995.
[35] C.-Y. Chen, C.-C. Chang, and W.-P. Yang, "Hybrid Method for Modular Exponentiation with Precomputation," Electronics Letters, vol. 32, no. 6, pp. 540-541, Mar. 1996.
[36] C. Doche, D. Kohel, and F. Sica, "Double-Base Number System for Multiscalar Multiplications," Proc. Int'l Conf. Theory and Applications of Cryptographic Techniques (EuroCrypt), pp. 502-519, 2009.
[37] C. Doche and L. Habsieger, "A Tree-Based Approach for Computing Double-Base Chains," Proc. 13th Australasian Conf. Information Security and Privacy (ACISP), pp. 433-446, 2008.
[38] J. Adikari, V.S. Dimitrov, and L. Imbert, "Hybrid Binary-Ternary Number System for Elliptic Curve Cryptosystems," IEEE Trans. Computers, 2010.
[39] K.W. Wong, E.C.W. Lee, L. Cheng, and X. Liao, "Fast Elliptic Scalar Multiplication Using New Double-Base Chain and Point Halving," Applied Math. and Computation, vol. 183, no. 2, pp. 1000-1007, 2006.
[40] C. Zhao, F. Zhang, and J. Huang, "Efficient Tate Pairing Computation Using Double-Base Chains," Science in China Series F: Information Sciences, vol. 51, no. 8, pp. 1096-1105, 2008.
[41] D.J. Bernstein and T. Lange, "Analysis and Optimization of Elliptic-Curve Single-Scalar Multiplication," Finite Fields and Applications, Contemporary Math., vol. 461, pp. 1-19, 2008.
[42] R. Bernstein, "Multiplication by Integer Constants," Software: Practice and Experience, vol. 16, no. 7, pp. 641-652, July 1986.
[43] J.D. Ullman, Computational Aspects of VLSI. W. H. Freeman & Co., 1984.
[44] I. Wegener and R. Pruim, Complexity Theory: Exploring the Limits of Efficient Algorithms. Springer-Verlag 2005.
[45] W.J. Paul and P.-M. Seidel, "To Booth or Not to Booth," The VLSI J., vol. 32, nos. 1/2, pp. 5-40, 2002.
[46] P. Kornerup, "Digit-Set Conversions: Generalizations and Applications," IEEE Trans. Computers, vol. 43, no. 5, pp. 622-629, May 1994.
[47] D.W. Matula, "Basic Digit Sets for Radix Representation," J. ACM, vol. 29, no. 4, pp. 1131-1143, 1982.
[48] P.-M. Seidel, L. McFearin, and D. Matula, "Secondary Radix Recodings for Higher Radix Multipliers," IEEE Trans. Computers, vol. 54, no. 2, pp. 111-123, Feb. 2005.
[49] D. Matula and P. Kornerup, "Finite Precision Rational Arithmetic: Slash Number Systems," IEEE Trans. Computers, vol. 34, no. 1, pp. 3-18, Jan. 1985.
[50] V. Oklobdzija, D. Villeger, and S. Liu, "A Method for Speed Optimized Partial Product Reduction and Generation of Fast Parallel Multipliers Using an Algorithmic Approach," IEEE Trans. Computers, vol. 45, no. 3, pp. 294-306, Mar. 1996.
[51] R. Zimmermann, "Datapath Synthesis for Standard-Cell Design," Proc. 19th IEEE Symp. Computer Arithmetic (ARITH), pp. 207-211, 2009.
[52] R. Zimmermann and D. Tran, "Optimized Synthesis of Sum-of-Products," Proc. Conf. Record of the 37th Asilomar Conf. Signals, Systems and Computers, vol. 1, pp. 867-872, 2003.
[53] A.D. Booth, "A Signed Binary Multiplication Technique," The Quarterly J. Mechanics and Applied Math., vol. 4, no. 2, pp. 236-240, 1951.
[54] A. Avizienis, "Signed-Digit Number Representations for Fast Parallel Arithmetic," IEEE Trans. Electronic Computers, vol. EC-10, no. 3, pp. 389-400, Sept. 1961.
[55] C.S. Wallace, "A Suggestion for a Fast Multiplier," IEEE Trans. Electronic Computers, vol. EC-13, no. 1, pp. 14-17, Feb. 1964.
[56] L. Dadda, "Some Schemes for Parallel Multipliers," Alta Frequenza, vol. 34, pp. 349-356, 1965.
[57] ARITH Research Group, "Arithmetic Module Generator Based on ARITH," Apr. 2010.
[58] Synopsys Inc., "Design Compiler Ultra," Apr. 2010.
[59] Synopsys Inc., "DesignWare IP," Apr. 2010.
23 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool