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New Multipliers Modulo 2/sup N/-1
August 1992 (vol. 41 no. 8)
pp. 957-961

Techniques for computing the product of two N-bit integers modulo 2/sup N/-1 from their k-bit byte decompositions are presented. A modulus 2/sup N/-1 is chosen, as multiplication performed in this modulus can be reconstructed from the cyclic convolution between the sequences of the k-bit bytes of the decomposed numbers. It is shown that cyclic convolutions can be computed using only additions and squaring operations but not two-operand multiplications. Since the squaring operation is a one-operand operation, significant savings in ROM bits can be obtained if look-up tables are used.

[1] A. V. Oppenheim and R. W. Schafer,Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975.
[2] F. J. Taylor, "Residue arithmetic: A tutorial with examples,"IEEE Comput. Mag., vol. 17, no. 5, pp. 50-62, May 1984.
[3] M. A. Soderstrand, W. K. Jenkins, G. A. Jullien, and F. J. Taylor, Eds.,Modern Applications of Residue Number System Arithmetic to Digital Signal Processing. New York: IEEE Press, 1986.
[4] F. J. Taylor, "Large moduli multipliers for signal processing,"IEEE Trans. Circuits Syst., vol. CAS-28, no. 7, pp. 731-736, July 1981.
[5] K. Hwang,Computer Arithmetic. New York: Wiley, 1979.
[6] M. J. Flynn and S. Waser,Introduction to Arithmetic for Digital Systems Designers. CBS College Publishing, 1982, pp. 215-222.
[7] A. Skavantzos, "The one over eight squared algorithm: A new way for computing convolutions and complex multiplications," inProc. Int. Symp. Circuits Syst.(ISCAS '90), New Orleans, LA, May 1990, pp. 61-64.
[8] A. Skavantzos, "Novel approach for implementing convolutions with small tables,"IEE Proc-E, vol. 138, no. 4, pp. 255-259, July 1991.
[9] J.H. McClellan and C.M. Rader,Number Theory in Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall, 1975.
[10] M.A. Soderstrand and E.L. Fields, "Multipliers for residue number arithmetic digital filters,"Electron. Lett., vol. 13, no. 6, pp. 164-166, Mar. 1977.
[11] M.A. Soderstrand and C. Vernia, "A High-speed low-cost moduloPimultiplier with RNS arithmetic applications," inProc. IEEE, vol. 68, no. 4, pp. 529-532, Apr. 1980.

Index Terms:
multipliers; modulo 2/sup N/-1; multiplication; cyclic convolution; additions; squaring; ROM bits; look-up tables; digital arithmetic; multiplying circuits.
Citation:
A. Skavantzos, P.B. Rao, "New Multipliers Modulo 2/sup N/-1," IEEE Transactions on Computers, vol. 41, no. 8, pp. 957-961, Aug. 1992, doi:10.1109/12.156538
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