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http://dx.doi.org/10.13067/JKIECS.2017.12.6.1181

On Efficient Algorithms for Generating Fundamental Units and their H/W Implementations over Number Fields  

Kim, Yong-Tae (Dept. of Mathematics Education, Gwangju National University of Education)
Publication Information
The Journal of the Korea institute of electronic communication sciences / v.12, no.6, 2017 , pp. 1181-1188 More about this Journal
Abstract
The unit and fundamental units of number fields are important to number field sieves testing primality of more than 400 digits integers and number field seive factoring the number in RSA cryptosystem, and multiplication of ideals and counting class number of the number field in imaginary quadratic cryptosystem. To minimize the time and space in H/W implementation of cryptosystems using fundamental units, in this paper, we introduce the Dirichlet's unit Theorem and propose our process of generating the fundamental units of the number field. And then we present the algorithm generating our fundamental units of the number field to minimize the time and space in H/W implementation and implementation results using the algorithm over the number field.
Keywords
Number Field; Unit; Dirchlet's Unit Theorem; Fundamental Units;
Citations & Related Records
Times Cited By KSCI : 3  (Citation Analysis)
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1 P. Dirichlet, Vorlesungen uber Zahlentheorie. Berlin: Springer Vieweg, 1894.
2 Z. Borevich and I. Shafarevich, Number Theory. New York: Translated by Newcomb Greenleaf for Scripta Technica, Academic Press, 1966.
3 M. Pohst and H. Zassenhaus, Algorithmic algebraic number theory. Cambridge: Cambridge University Press, 2002.
4 E. Fouvry and J. Kluners, "On the negative Pell equation," Annals of Mathematics, vol. 172, no. 3, 2010, pp. 235-254.
5 P. Stevenhagen, Number rings. Leiden,: Universiteit Leiden Press, 2017.
6 J. Lee and S. Louboutinb, "Determination of the orders generated by a cyclic cubic unit that are Galois invariant," J. of Number Theory, vol. 148, no. 1, 2015, pp. 33-39.   DOI
7 K. Wang, "Fundamental unit system and class number of real bicyclic biquadratic number fields," Proc. of the Japan Academy, Ser. A, Mathematical Sciences, Tokyo, Japan, vol. 77, no. 9, May, 2001, pp. 147-150.   DOI
8 H. Kim, S. Cho, U. Choi, M. Kwon, and G. Kong, "Synthesis of Uniform CA and 90/150 Hybrid CA" J. of the Korea Institute of Electronic Communication Sciences, vol. 11, no. 3, 2016, pp. 293-302.   DOI
9 U. Choi, S. Cho, H. Kim, M. Kwon, and S. Kim, "Synthesis of 90/102(170)/150 linear CA using 90/150 linear CA," J. of the Korea Institute of Electronic Communication Sciences, vol. 11, no. 9, 2016, pp. 885-891.   DOI
10 N. Tschebotareff, "Die Bestimmung der Dichtigkeit einer Menger von Primzahlen, welsche zu einer gegebenen Substitutions-klasse gehoren," Mathematische Annalen, vol. 95, no. 1, 1926, pp. 191-228.   DOI
11 H. Kim, S. Cho, M. Kwon, and H. An, "A study on the cross sequences," J. of the Korea Institute of Electronic Communication Sciences, vol. 7, no. 1, 2012, pp. 61-67.   DOI
12 S. Wolfram, Mathematica. 4th Ed.. New York: Wolfram Champaign Research, Inc., 1999.