This Article 
 Bibliographic References 
 Add to: 
Closed-Form Expression for the Average Weight of Signed-Digit Representations
August 1999 (vol. 48 no. 8)
pp. 848-851

Abstract—In radix-$r$ number system, the minimal weight signed-digit (SD) representation has minimal number of nonzero signed-digits which belong to the set $\{\pm{1},\pm{2},\ldots,\pm{(r-1)}\}$. In this article, we derive closed form expressions for the average number of nonzero digits in the minimal weight SD representation and for the average length of the canonical SD representation, a special case of the minimal weight SD form, of a positive integer whose radix-$r$ form is of length $\schmi{n}$, $\schmi{n}\geq 1$.

[1] S. Arno and F.S. Wheeler, Signed Digit Representations of Minimal Hamming Weight IEEE Trans. Computers, vol. 42, no. 8, pp. 1007-1010, Aug. 1993.
[2] A. Avizienis, “Signed-Digit Number Representations for Fast Parallel Arithmetic,” IRE Trans. Electronic Computers, vol. 10, pp. 389-400, 1961.
[3] E.F. Brickell, D.M. Gordon, K.S. McCurley, and D.B. Wilson, “Fast Exponentiation with Precomputation (Extended Abstract),” Proc. EUROCRYPT '92, pp. 200-207, 1992.
[4] W.E. Clark and J.J. Liang, On Arithmetic Weight for a General Radix Representation of Integers IEEE Trans. Information Theory, vol. 19, no. 6, pp. 823-826, 1973.
[5] O. Egecioglu and C.K. Koc, “Fast Modular Exponentiation” Proc. 1990 Bilkent Int'l Conf. New Trends in Comm., Control, and Signal Processing, Ankara, pp. 188-194, 1990.
[6] D. Gollmann, Y. Han, and C.J. Mitchell, “Redundant Integer Representations and Fast Exponentiation” Designs, Codes, and Cryptography, vol. 7, pp. 135-151, 1996.
[7] K. Hwang, Computer Arithmetic. New York: Wiley, 1979.
[8] D. Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, Reading, Mass., 1998.
[9] F. Morain and J. Olivos, “Speeding Up the Computations on an Elliptic Curve Using Addition-Subtraction Chains” Theoretical Informatics and Applications, vol. 24, pp. 531-543, 1990.
[10] G.W. Reitwiesner, “Binary Arithmetic,” Advan. Computers 1, pp. 232-308. Academic Press, 1960.
[11] H. Wu and M.A. Hasan, Efficient Exponentiation of a Primitive Root in$GF(2^m)$ IEEE Trans. Computers, vol. 46, no. 2, pp. 162-172, Feb. 1997.

Index Terms:
Radix-$r$ number system, minimal weight signed-digit representation, canonical signed-digit representation.
Huapeng Wu, M. Anwar Hasan, "Closed-Form Expression for the Average Weight of Signed-Digit Representations," IEEE Transactions on Computers, vol. 48, no. 8, pp. 848-851, Aug. 1999, doi:10.1109/12.795126
Usage of this product signifies your acceptance of the Terms of Use.