MODULAR EXPONENTIATION
DOI:
https://doi.org/10.47839/ijc.2.3.229Keywords:
Modular exponentiation, basic techniques, fixed-exponent techniques, fixed-base techniques, techniques based on modulus particularities, binary fan-in techniqueAbstract
Exponentiation is a fundamental operation in computational number theory. Primality testing and cryptography are important working fields in which the exponentiation is heavily used. In this paper we survey the most popular methods for modular exponentiation: basic techniques, fixed-exponent techniques, fixed-base techniques, and techniques based on modulus particularities. Some aspects related to parallelism are also discussed.References
S. Iftene. Modular exponentiation. Pre- Proceedings of the NATO Advanced Research Workshop “Concurent Information Proccesing and Computing”, July 5-10, 2003, Sinaia, Romania, pp. 125-144
F.L. Tiplea, S. Iftene, C. Hritcu, I. Goriac, R.M. Gordan and E. Erbiceanu. MpNT: A Multi-Precision Number Theory Package. Number-Theoretic Algorithms (I). Technical Report TR03-02 (2003), Faculty of Computer Science “Al.I.Cuza” University of Iasi. http://www.infoiasi.ro/~tr/tr.pl.cgi
E.G. Thurber. On addition chains l(mn) ? l(n)-b and lower bounds for c(r), Duke Mathematical Journal 40 (1973), pp. 907-913
B. Moller. Improved Techniques for Fast Exponentiation. Proceedings of the Workshop “Information Security and Cryptology ICISC 2002”, pp. 298-312
A. Brauer. On Addition Chains, Bulletin of the American Mathematical Society 45 (1939), pp. 736-739
A. C. Yao. On the evaluation of powers, SIAM Journal on Computing 5 (1976), pp. 100-103
D.E. Knuth. The Art of Computer Programming. Seminumerical Algorithms. Addison-Wesley, third edition, 1997
D.E. Knuth. The Art of Computer Programming. Seminumerical Algorithms. Addison-Wesley, second edition, 1981
D.R. Stinson. Some observations on parallel algorithms for fast exponentiation in GF(2n). SIAM Journal on Computing 19 (4) (1990), pp. 711-717
M. Nocker. Some remarks on parallel exponentiation. Extended abstract. Proceedings of the Workshop “International Symposium on Symbolic and Algebraic Computation ISSAC 2000”, pp. 250-257
E. F. Brickell, D. M. Gordon, K. S. McCurley and D. B. Wilson. Fast Exponentiation with Precomputation: Algorithms and Lower Bounds,preprint,1995. http://research.microsoft.com/~dbwilson/bgmw/
P. De Rooij. Efficient exponentiation using precomputation and vector addition chains. Proceedings of the Workshop “EUROCRYPT 1994”, pp. 389-399
C. H. Lim and P.J. Lee. More flexible exponentiation with precomputation. Proceedings of the Workshop“CRYPTO 1994”, pp. 95-107
D. J. Bernstein. Pippenger's exponentiation algorithm, preprint, 1991. http://cr.yp.to/papers.html
R. L. Rivest, A. Shamir and L. M. Adelman. A Method for Obtaining Digital Signatures and Public-key Cryptosystems. Communications of the ACM 2 (21) (1978), pp. 120-126.
T. Collins, D. Hopkins, S. Langford and M. Sabin. Public Key Cryptographic Apparatus and Method. United States Patent 5, 848, 159 (1997)
H. Garner. The residue number system. IRE Transactions on Electronic Computers EC-8 (1959), pp. 140-147.
J.-J. Quisquater and C. Couvreur. Fast decipherment algorithm for the RSA public-key cryptosystem. IEE Electronics Letters 8 (21) (1982), pp. 905-907.
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