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Optimal Left-to-Right Binary Signed-Digit Recoding
July 2000 (vol. 49 no. 7)
pp. 740-748

Abstract—This paper describes new methods for producing optimal binary signed-digit representations. This can be useful in the fast computation of exponentiations. Contrary to existing algorithms, the digits are scanned from left to right (i.e., from the most significant position to the least significant position). This may lead to better performances in both hardware and software.

[1] A. Avizienis, “Signed-Digit Number Representations for Fast Parallel Arithmetic,” IRE Trans. Electronic Computers, vol. 10, pp. 389-400, 1961.
[2] I. Koren, Computer Arithmetic Algorithms.Englewood Cliffs, N.J.: Prentice Hall, 1993.
[3] N. Takagi,H. Yasuura,, and S. Yajima,“High-speed VLSI multiplication algorithm with a redundant binary addition tree,” IEEE Trans. Computers, vol. 34, no. 9, pp. 789-796, Sept. 1985.
[4] S. Kuninobu, T. Nishiyama, H. Edamatsu, T. Taniguchi, and N. Takagi, “Design of High Speed MOS Multiplier and Divider Using Redundant Binary Representation,” Proc. Eighth Symp. Computer Arithmetic, pp. 80-86, 1987.
[5] Y. Harata, Y. Nakamura, H. Nagese, M. Takigawa, and N. Takagi, "A High-Speed Multiplier Using a Redundant Binary Adder Tree," IEEE J. Solid-State Circuits, vol. 22, pp. 28-34, Feb. 1987.
[6] A. Vandemeulebroecke,E. Vanzieleghem,T. Denayer, and P.G.A. Jespers,"A New Carry-Free Diversion Algorithm and Its Application to a Single-Chip 1024-b RSA Processor," IEEE J. Solid State Circuits, vol. 25, no. 3, pp. 748-765, 1990.
[7] K. Hwang,Computer Arithmetic, Principles, Architecture, and Design.New York: John Wiley&Sons, 1979.
[8] S.M. Yen, C.S. Laih, C.H. Chen, and J.Y. Lee, “An Efficient Redundant-Binary Number to Binary Number Converter,” IEEE J. Solid-State Circuits, vol. 27, no. 1, pp. 109-112, 1992.
[9] 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.
[10] Ç.K. Koç, “High-Speed RSA Implementations,” Technical Report TR 201, RSA Laboratories, Nov. 1994.
[11] D. Knuth, The Art of Computer Programming, Vol. 2, Addison-Wesley, Reading, Mass., 1998.
[12] P. Downey, B. Leong, and R. Sethi, “Computing Sequences with Addition Chains,” SIAM J. Computing, vol. 10, pp. 638-646, 1981.
[13] J. Bos and M. Coster, "Addition Chain Heuristics," Proc. Crypto '89, Lecture Notes in Computer Science, vol. 435, pp. 400-407. Springer-Verlag, 1990.
[14] T. ElGamal, A Public-Key Cryptosystem and a Signature Scheme Based on Discrete Logarithms IEEE Trans. Information Theory, vol. 31, no. 4, pp. 469-472, 1985.
[15] A.J. Menezes, P.C. van Oorschot, and S.A. Vanstone, Handbook of Applied Cryptography, CRC Press, Boca Raton, Fla., 1996, pp. 543-590.
[16] E.F. Brickell, D.M. Gordon, K.S. McCurley, and D.R. Wilson, “Fast Exponentiation with Precomputation: Algorithms and Lower Bounds,” preprint, Mar. 1995. An earlier version appeared in Proc. EUROCRYPT '92.
[17] 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.
[18] A.D. Booth, “A Signed Binary Multiplication Technique,” Quarterly J. Mechanics and Applied Math., vol. 4, pp. 236-240, 1951.
[19] G.W. Reitwiesner, “Binary Arithmetic,” Advances in Computers, vol. 1, pp. 231-308, 1960.
[20] J. Jedwab and C.J. Mitchell, Minimum Weight Modified Signed-Digit Representations and Fast Exponentiation Electronics Letters, vol. 25, no. 17, pp. 1171-1172, 1989.
[21] H. Cohen, A Course in Computational Algebraic Number Theory. Springer-Verlag, 1993.
[22] Ö. Egecioglu and Ç. K. Koç, "Exponentiation Using Canonical Recoding," Theoretical Computer Science, vol. 129, no. 2, pp. 407-417, 1994.
[23] B.S. Kaliski Jr., “The Montgomery Inverse and Its Applications,” IEEE Trans. Computers, vol. 44, no. 8, pp. 1,064-1,065, Aug. 1995.
[24] 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.
[25] D.M. Gordon, “A Survey of Fast Exponentiation Methods” J. Algorithms, vol. 27, no. 1, pp. 129-146, Apr. 1998.
[26] 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.
[27] 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.
[28] J. Omura and J. Massey, “Computational Method and Apparatus for Finite Field Arithmetic,” U.S. Patent #4,587,627, 1986.
[29] Ç.K. Koç, “Parallel Canonical Recoding,” Electronics Letters, vol. 32, pp. 2,063-2,065, 1996.
[30] Computer Arithmetic, E.E. Swartzlander Jr., ed., vols. 1 and 2. IEEE CS Press, 1990.

Index Terms:
Computer arithmetic, converter, signed-digit representation, redundant number representation, SD2 left-to-right recoding, canonical/minimum-weight/nonadjacent form, exponentiation, elliptic curves, smart-cards, cryptography.
Marc Joye, Sung-Ming Yen, "Optimal Left-to-Right Binary Signed-Digit Recoding," IEEE Transactions on Computers, vol. 49, no. 7, pp. 740-748, July 2000, doi:10.1109/12.863044
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