This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Closed-Form Expression for the Average Weight of Signed-Digit Representations
August 1999 (vol. 48 no. 8)
pp. 848-851
ASCII Text
x 
 
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, August, 1999.
 
 
BibTex
x
 
@article{ 10.1109/12.795126,
author = {Huapeng Wu and M. Anwar Hasan},
title = {Closed-Form Expression for the Average Weight of Signed-Digit Representations},
journal ={IEEE Transactions on Computers},
volume = {48},
number = {8},
issn = {0018-9340},
year = {1999},
pages = {848-851},
doi = {http://doi.ieeecomputersociety.org/10.1109/12.795126},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Computers
TI - Closed-Form Expression for the Average Weight of Signed-Digit Representations
IS - 8
SN - 0018-9340
SP848
EP851
EPD - 848-851
A1 - Huapeng Wu,
A1 - M. Anwar Hasan,
PY - 1999
KW - Radix-$r$ number system
KW - minimal weight signed-digit representation
KW - canonical signed-digit representation.
VL - 48
JA - IEEE Transactions on Computers
ER -
 

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.
Citation:
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.