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Theory and Applications of the Double-Base Number System
October 1999 (vol. 48 no. 10)
pp. 1098-1106

Abstract—In this paper, we analyze some of the main properties of a double base number system, using bases 2 and 3; in particular, we emphasize the sparseness of the representation. A simple geometric interpretation allows an efficient implementation of the basic arithmetic operations and we introduce an index calculus for logarithmic-like arithmetic with considerable hardware reductions in look-up table size. We discuss the application of this number system in the area of digital signal processing; we illustrate the discussion with examples of finite impulse response filtering.

[1] A. Borodin and P. Towari, “On the Decidability of Sparse Univariate Polynomial Interpolation,” Computational Complexity, vol. 1, pp. 67-90, 1991.
[2] P. Montgomery, “A Survey of Modern Integer Factorization Algorithms,” CWI Quarterly, vol. 7, no. 4, pp. 337-366, 1994.
[3] M.D. Ercegovac, T. Lang, J.G. Nash, and L.P. Chow, “An Area-Time Efficient Binary Divider,” IEEE Int'l Conf. Computer Design, pp. 645-648, Rye Brook, N.Y., Oct. 1987.
[4] P. Kornerup, “Computer Arithmetic: Exploiting Redundancy in Number Representations,” Proc. ASAP '95, Strasbourg, France, 1995.
[5] A. Avizienis, “Signed-Digit Number Representation for Fast Parallel Arithmetic,” IRE Trans. Electronic Computers, vol. 10, pp. 389-400, 1961.
[6] H. Garner, “Number Systems and Arithmetic,” Advances in Computing, vol. 6, pp. 131-194, 1965.
[7] G.W. Reitwiesner, “Binary Arithmetic,” Advances in Computing, vol. 1, pp. 231-308, 1960.
[8] E. Swartzlander, “Digital Optical Computing,” Applied Optics, vol. 25, pp. 3,021-3,032, 1986.
[9] 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.
[10] T.N. Shorey and R. Tijdeman, Exponential Diophantine Equations. Cambridge Univ. Press, 1986.
[11] G. Hardy, Ramanujan. Cambridge Univ. Press, 1940.
[12] B.M.M. de-Weger, “Algorithms for Diophantine Equations,” CWI Tracts-Amsterdam, vol. 65, 1989.
[13] S.S. Pillai, “On the Equation$2^a-2^b=3^c-3^d$,” Bulletin of the Calcutta Math. Soc., vol. 37, pp. 15-20, 1945.
[14] A. Baker, “The Theory of Linear Forms in Logarithms,” Transcendental Theory—Advances and Applications, A. Baker, ed., pp. 1-27, Academic Press, 1987.
[15] S. Sadeghi-Emamchaie, G.A. Jullien, V. Dimitrov, and W.C. Miller, “Digital Arithmetic Using Analog Arrays,” Proc. Eighth Great Lakes Symp. VLSI, pp. 202-207, Lafayette, La., Feb. 1998.
[16] E.E. Swartzlander, D.V. Chandra, H.T. Nagel, and S.A. Starks, “Sign/Logarithmic for FFT Implementation,” IEEE Trans. Computers, vol. 32, pp. 526-534, 1983.
[17] D. Lewis, “An Accurate LNS Arithmetic Unit Using Interleaved Memory Function Interpolator,” Proc. ARITH-11, pp. 2-9, Windsor, Ontario, Canada, 1993.
[18] V.S. Dimitrov and T.V. Cooklev, “Two Algorithms for Modular Exponentiation Using Nonstandard Arithmetic,” IEICE Trans. Fundamentals, pp. 82-87, 1995.
[19] A.J. Brentjes, “Multi-Dimensional Continued Fraction Algorithms,” Mathematical Centre Tracts, Amsterdam, vol. 145, 1981.
[20] C.-Y. Chen, C.-C. Chang, and W.-P. Yang, “Hybrid Method for Modular Exponentiation with Precomputations,” IEE Electronics Letters, vol. 32, no. 6, pp. 540-541, 1996.
[21] H.R.P. Ferguson and R.W. Forcade, “Generalization of the Euclidean Algorithm for Real Numbers for All Dimensions Higher than Two,” Bulletin Am. Math. Soc., vol. 1, pp. 912-914, 1979.
[22] R.J. Stroeker and R. Tijdeman, “Diophantine Equations,” Computational Methods in Number Theory, H. Lenstra and R. Tijdeman, eds., Math. Centre Tracts-Amsterdam, vol. 155, pp. 321-369, 1987.
[23] P. Erdos, “Quickies,” Math. Magazine, vol. 66, p. 67, 1994.
[24] P. Erdos and M. Lewin, “d-Complete Sequences of Integers,” Math. of Computation, vol. 65, pp. 837-840, 1996.
[25] G. Dombi and B. Valko, “On a Problem of Erdos,” Acta Mathematica Hungarica, vol. 77, pp. 47-56, 1997.
[26] R. Blecksmith, M. McCallum, and J.L. Selfridge, “3-Smooth Representation of Integers,” Am. Math. Monthly, vol. 105, pp. 529-543, 1998.

Index Terms:
Double-base number system, index calculus, digital signal processing, FIR filters.
Vassil S. Dimitrov, Graham A. Jullien, William C. Miller, "Theory and Applications of the Double-Base Number System," IEEE Transactions on Computers, vol. 48, no. 10, pp. 1098-1106, Oct. 1999, doi:10.1109/12.805158
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