Exponentiation using division chains

IEEE Transactions on Computers

Volumn 47, Issue 7, 1998, Pages 757-765

  Walter, Colin D ^a,b  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

^b IEEE   (United Kingdom)

Author keywords

Addition chains; Bit recoding; m ary method; Mixed basis arithmetic; Modular exponentiation; Radix representation; RSA cryptosystem

Indexed keywords

ALGORITHMS; CALCULATIONS; CRYPTOGRAPHY; NUMBER THEORY;

ADDITION CHAIN; BIT RECODING; DIVISION CHAIN; EXPONENTIATION; M-ARY METHOD; MIXED BASIS ARITHMETIC; RADIX REPRESENTATION; RSA CRYPTOGRAPHY;

DIGITAL ARITHMETIC;

EID: 0032118606     PISSN: 00189340     EISSN: None     Source Type: Journal    
DOI: 10.1109/12.709375     Document Type: Article

References (26)

1. I.E. Bocharova and B.D. Kudryashov, "Fast Exponentiation in Cryptography," Lecture Notes in Computer Science, vol. 948, pp. 146-157. Springer-Verlag, 1995.
2. J. Bos and M. Coster, "Addition Chain Heuristics," Proc. Crypto '89, Lecture Notes in Computer Science, vol. 435, pp. 400-407. Springer-Verlag, 1990.
3. E.F. Brickell, D.M. Gordon, K.S. McCurley, and D.B. Wilson, "Fast Exponentiation with Precomputation," Pros. Eurocrypt '92, Lecture Notes in Computer Science, vol. 658, pp. 200-207. Springer-Verlag, 1993.
4. C.Y. Chen, C.C. Chang, and W.P. Yang, "Hybrid Method for Modular Exponentiation with Precomputation," Electronics Letters, vol. 32, no. 6, pp. 540-541, 1996.
5. L.S. Danilchenko, "Efficient Algorithms for Remainder Computation and Exponentiation of Long Numbers," Cybernetics and Systems Analysis, vol. 32, no. 3, pp. 437-441, 1996.
6. V. Dimitrov and T. Cooklev, "Two Algorithms for Modular Exponentiation Using Non-Standard Arithmetics," IEICE Trans. Fundamentals of Electronics, Comm., and Computer Sciences, vol. E78-A, no. 1, pp. 82-87, Jan. 1995.
7. V.S. Dimitrov, G.A. Jullien, and W.C. Miller, "Theory and Applications for a Double-Base Number System," Proc. 13th IEEE Symp. Computer Arithmetic, pp. 44-51, Monterey, Calif., 6-9 July 1997.
8. P. Downey, B. Leong, and R. Sethi, "Computing Sequences with Addition Chains," SIAM J. Computing, vol. 10, no. 3, pp. 638-646, 1981.
9. Ö. Egecioglu and Ç. K. Koç, "Fast Modular Exponentiation," Comm., Contro,l and Signal Processing, E. Arikan, ed., pp. 188-194. Elsevier Science, 1990.
10. Ö. Egecioglu and Ç. K. Koç, "Exponentiation Using Canonical Receding," Theoretical Computer Science, vol. 129, no. 2, pp. 407-417, 1994.
11. P. Erdös, "Remarks on Number Theory III: On Addition Chains," Acta Arithmetica, vol. 6, pp. 77-81, 1960.
12. L.C.K. Hui and K.-Y. Lam, "Fast Square-and-Multiply Exponentiation for RSA," Electronics Letters, vol. 30, no. 17, pp. 1,396-1,397, Aug. 1994.
13. S. Kawamura, K. Takabayashi, and A. Shimbo, "A Fast Modular Exponentiation Algorithm," IEICE Trans. Comm., Electronics, Information, and Systems, vol. E-74, no. 8, pp. 2,136-2,142, Aug. 1991.
14. D.E. Knuth, The Art of Computer Programming, vol. 2, "Seminumerical Algorithms," second ed., section 4.6.3, pp. 441-466. Addison-Wesley, 1981.
15. Ç.K. Koç, "High Radix and Bit Recoding Techniques for Modular Exponentiation," Int'l J. Computer Mathematics, vol. 40, nos. 3-4, pp. 139-156, 1991.
16. Ç.K. Koç, "Analysis of Sliding Window Techniques for Exponentiation," Computers & Mathematics with Applications, vol. 30, no. 10, pp. 17-24, 1995.
17. D.C. Lou and C.C. Chang, "Fast Exponentiation Method Obtained by Folding the Exponent in Half," Electronics Letters, vol. 32, no. 11, pp. 984-985, 1996.
18. J. Olivos, "On Vectorial Addition Chains," J. Algorithms, vol. 2, no. 1, pp. 13-21, 1981.
19. P. de Rooij, "Efficient Exponentiation using Precomputation and Vector Addition Chains," Proc. Eurocrypt '94, Lecture Notes in Computer Science, vol. 950, pp. 389-399. Springer-Verlag, 1995.
20. Y. Tsuruoka and K. Koyama, "Fast Exponentiation Algorithms Based on Batch-Processing and Precomputation," IEICE Trans. Fundamentals of Electronics, Comm., and Computer Sciences, vol. E80-A, no. 1, pp. 34-39, Jan. 1997.
21. C.D. Walter, "Space/Time Trade-Offs for Higher Radix Modular Multiplication Using Repeated Addition," IEEE Trans. Computers, vol. 46, no. 2, pp. 139-141, Feb. 1997.
22. Y. Yacobi, "Exponentiating Faster with Addition Chains," Advances in Cryptology - Eurocrypt '90, Lecture Notes in Computer Science, vol. 473, pp. 222-229. Springer-Verlag, 1991.
23. A. Yao, "On the Evaluation of Powers," SIAM J. Computing, vol. 5, pp. 100-103, 1976.
24. S.M. Yen, "Improved Common-Multiplicand Multiplication and Fast Exponentiation by Exponent Decomposition," IEICE Trans. Fundamentals of Electronics Comm. and Computer Sciences, vol. E80-A, no. 6, pp. 1,160-1,163, 1997.
25. S.M. Yen and C.-S. Laih, "Common-Multiplicand Multiplication and its Applications to Public Key Cryptography," Electronics Letters, vol. 29, no. 17, pp. 1,583-1,584, 1993.
26. C.N. Zhang, H.L. Martin, and D.Y.Y. Yun, "Parallel Algorithms and Systolic Array Designs for RSA Cryptosystem," Proc. Int'l Conf. Systolic Arrays, K. Bromley, S.Y. Kung, and E. Swartzlander, eds., pp. 341-350, San Diego, Calif., 25-27 May 1988.