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http://dx.doi.org/10.9708/jksci.2013.18.8.087

Modified Baby-Step Giant-Step Algorithm for Discrete Logarithm  

Lee, Sang-Un (Dept. of Multimedia Eng., Gangneung-Wonju National University)
Abstract
A baby-step giant-step algorithm divides n by n blocks that possess $m={\lceil}\sqrt{n}{\rceil}$ elements, and subsequently computes and stores $a^x$ (mod n) for m elements in the 1st block. It then calculates mod n for m blocks and identifies each of them with those in the 1st block of an identical elemental value. This paper firstly proposes a modified baby-step giant-step algorithm that divides ${\lceil}m/2{\rceil}$ blocks with m elements applying $a^{{\phi}(n)/2}{\equiv}1(mod\;n)$ and $a^x(mod\;n){\equiv}a^{{\phi}(n)+x}$ (mod n) principles. This results in a 50% decrease in the process of the giant-step. It then suggests a reverse baby-step giant step algorithm that performs and saves ${\lceil}m/2{\rceil}$ blocks firstly and computes $a^x$ (mod n) for m elements. The proposed algorithm is found to successfully halve the memory and search time of the baby-step giant step algorithm.
Keywords
Discrete logarithm; Discrete exponentiation; Multiplicative order; Euler's totient function; Baby-step giant-step;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
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1 J. S. Coron, D. Lefranc, and G. Poupard, "A new baby-step giant-step algorithm and some applications to cryptanalysis," 7th International Workshop, Vol. 3659, pp. 47-60, Aug. 2005.
2 T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, "Introduction to Algorithms," 2nd Ed., MIT Press and McGraw-Hill. pp. 887-896, 2001.
3 Wikipedia, "Discrete Logarithm," http://en. wikipedia.org/wiki/Discrete_logarithm, 2013.
4 D. R. Stinson, "Cryptography: Theory and Practice," 3rd ed., London, CRC Press, 2006.
5 A. Stein and E. Teske, "Optimized Baby step- Giant step Methods," Journal of the Ramanujan Mathematical Society, Vol. 20, No. 1, pp. 1-32, Jan. 2005.
6 D. C. Terr, "A modification of Shanks' Baby-step Giant-step algorithm," Mathematics of Computation, Vol. 69, No. 230, pp. 767-773, Apr. 2000.
7 B. K. Oh, K. C. Ha, and J. H. Oh, "An Improved Baby-Step-Giant-Step Method For Certain Elliptic Curves," Journal of Applied Mathematics & Computing, Vol. 20, No. 1-2, pp. 485-489, 2006.
8 S. U. Lee, "Square-and-Divide Modular Exponentiation," Journal of Korean Society of Computer Information, Vol. 18, No. 4, pp. 123-129, Apr. 2013.   과학기술학회마을   DOI   ScienceOn