Abstract
An efficient algorithm for exponentiation with precomputation is proposed and analyzed. Due to the precomputations, it is mainly useful for (discrete log based) systems with a fixed base.
The algorithm is an instance of a more general method. This method is based on two ideas. Firstly, the idea from [3] of splitting the exponentiation into the product of a number of exponentiations with smaller exponents. Secondly, the use of the technique of vector addition chains to compute this product of powers.
Depending on the amount of precomputations and memory, the proposed algorithm is about two to six times as fast as binary square and multiply. It is slightly slower than the method from [3], but requires far less memory.
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References
J. Bos and M. Coster, “Addition chain heuristics”, Advances in Cryptology — Proceedings of Crypto'89 (G. Brassard, ed.), Lecture Notes in Computer Science, vol. 435, Springer-Verlag, 1990, pp. 400–407.
J. N. E. Bos, Practical Privacy, Ph.D. thesis, Technical University of Eindhoven, March 1992.
E. F. Brickell, D. M. Gordon, K. S. McCurley, and D. B. Wilson, “Fast exponentiation with precomputation (extended abstract)”, Advances in Cryptology — Proceedings of Eurocrypt'92 (R. A. Rueppel, ed.), Lecture Notes in Computer Science, vol. 658, Springer-Verlag, 1993, pp. 200–207.
E. F. Brickell and K. S. McCurley, “An interactive identification scheme based on discrete logarithms and factoring”, Journal of Cryptology 5 (1992), no. 1, pp. 29–39.
M. Coster, Some Algorithms on Addition Chains and their Complexity, Tech. Report CS-R9024, Centrum voor Wiskunde en Informatica, Amsterdam, 1990.
T. ElGamal, “A public key cryptosystem and a signature scheme based on discrete logarithms”, IEEE Transactions on Information Theory IT-31 (1985), no. 4, pp. 469–472.
D. E. Knuth, Seminumerical Algorithms, second ed., The Art of Computer Programming, vol. 2, Addison-Wesley, Reading, Massachusetts, 1981.
National Institute of Technology and Standards, Specifications for the Digital Signature Standard (DSS), Federal Information Processing Standards Publication XX, US. Department of Commerce, February 1 1993.
R. L. Rivest, A. Shamir, and L. Adleman, “A method for obtaining digital signatures and public-key cryptosystems”, Communications of the ACM 21 (1978), no. 2, pp. 120–126.
C. P. Schnorr, “Efficient signature generation by smart cards”, Journal of Cryptology 4 (1991), no. 3, pp. 161–174.
A. Yao, “On the evaluation of powers”, SIAM Journal on Computing 5 (1976), no. 1, pp. 100–103.
S.-M. Yen and C.-S. Laih, “The fast cascade exponentiation algorithm and its application to cryptography”, Abstracts of Auscrypt 92, 1992, pp. 10-20–10-25.
S.-M. Yen, C.-S. Laih, and A. K. Lenstra, “A note on multi-exponentiation”, IEE Proceedings, Computers and Digital Techniques 141 (1994), no. 5, to appear.
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© 1995 Springer-Verlag Berlin Heidelberg
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de Rooij, P. (1995). Efficient exponentiation using precomputation and vector addition chains. In: De Santis, A. (eds) Advances in Cryptology — EUROCRYPT'94. EUROCRYPT 1994. Lecture Notes in Computer Science, vol 950. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0053453
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DOI: https://doi.org/10.1007/BFb0053453
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