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Modulo Reduction in Residue Number Systems
May 1995 (vol. 6 no. 5)
pp. 449-454
ASCII Text
x 
 
Karl C. Posch, Reinhard Posch, "Modulo Reduction in Residue Number Systems," IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 5, pp. 449-454, May, 1995.
 
 
BibTex
x
 
@article{ 10.1109/71.382314,
author = {Karl C. Posch and Reinhard Posch},
title = {Modulo Reduction in Residue Number Systems},
journal ={IEEE Transactions on Parallel and Distributed Systems},
volume = {6},
number = {5},
issn = {1045-9219},
year = {1995},
pages = {449-454},
doi = {http://doi.ieeecomputersociety.org/10.1109/71.382314},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Parallel and Distributed Systems
TI - Modulo Reduction in Residue Number Systems
IS - 5
SN - 1045-9219
SP449
EP454
EPD - 449-454
A1 - Karl C. Posch,
A1 - Reinhard Posch,
PY - 1995
VL - 6
JA - IEEE Transactions on Parallel and Distributed Systems
ER -
 

Abstract—Residue number systems provide a good means for extremely long integer arithmetic. Their carry-free operations make parallel implementations feasible. Some applications involving very long integers, such as public key encryption, rely heavily on fast modulo reductions. This paper shows a new combination of residue number systems with efficient modulo reduction methods. Two methods are compared, and the faster one is scrutinized in detail. Both methods have the same order of complexity, $O(\log n)$, with $n$ denoting the amount of registers involved.

Index Terms—Computer arithmetic, cryptography, distributed systems, hardware algorithms, long integer arithmetic, modulo reduction, parallel algorithms, residue number systems.

[1] S. R. Duss\' eand B. S Kaliski Jr.,“A cryptographic library for the motorola DSP 56000,”inProc. Advances in Cryptology—Eurocrypt '90, New York, 1990.
[2] D. Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, Reading, Mass., 1998.
[3] H. L\" uneburg,Vorlesungen$\ddot{u}$ber Zahlentheorie, Elemente der Mathematik vom h\"{o}heren Standpunkt aus, Band VIII, E. Trost, Ed. Basel: Birkh\"{a}user Verlag, 1978.
[4] P. L. Montgomery,“Modular multiplication without trial division,”Mathemat. Comput., vol. 44, no. 170, pp. 519–521, Apr. 1985.
[5] K. C. Posch and R. Posch,“Approaching encryption at ISDN speed using partial parallel modulus multiplication,”Microprocessing and Microprogramming. Amsterdam, The Netherlands: North-Holland, 1990, vol. 29, pp. 177–184.
[6] ——,“Base extension using a convolution sum in residue number systems,”Computing 50. New York: Springer-Verlag, 1993, pp. 93–104.
[7] R.L. Rivest,A. Shamir, and L.A. Adleman,"A Method for Obtaining Digital Signatures and Public Key Cryptosystems," Comm. ACM, vol. 21, pp. 120-126, 1978.
[8] L. Shoenfeld,“Sharper bounds for the Chebyshev funtions$\Theta(x)$and$\Psi(x)$,”Math. Comp., vol. 30, pp. 337–360, 1976.
[9] A. P. Shenoy and R. Kumaresan,“Fast base extension using a redundant modulus in RNS,”IEEE Trans. Comput., vol. 38, pp. 292–297, Feb. 1989.
[10] J. Schwemmlein, R. Posch, and K. C. Posch,“High performance modular arithmetic using an RNS based Chipset,”inProc. Conf. Massively Parallel Comput. Syst.: The Challenges of General-Purpose and Special Purpose Comput., Ischia, Italy, 1994.
[11] N. S. Szabo and R. I. Tanaka,Residue Arithmetic and Its Applications to Computer Technology. New York: McGraw-Hill, 1967.
[12] F. J. Taylor,“Residue arithmetic: A tutorial with examples,”IEEE Comput. Mag., pp. 50–62, May 1984.

Citation:
Karl C. Posch, Reinhard Posch, "Modulo Reduction in Residue Number Systems," IEEE Transactions on Parallel and Distributed Systems, vol. 6, no. 5, pp. 449-454, May 1995, doi:10.1109/71.382314
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