• Bulletin of the Korean Mathematical Society
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• v.59 no.3
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• pp.609-615
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• 2022

We give a formula for the sizes of the dual groups. It is obtained by generalizing a size estimation of certain algebraic structure that lies in the heart of the proof of the celebrated primality test by Agrawal, Kayal and Saxena. In turn, by using our formula, we are able to give a streamlined survey of the AKS test.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.13 no.3
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• pp.103-109
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• 2013

Miller-Rabin method is the most prevalently used primality test. However, this method mistakenly reports a Carmichael number or semi-prime number as prime (strong lier) although they are composite numbers. To eradicate this problem, it selects k number of m, whose value satisfies the following : m=[2,n-1], (m,n)=1. The Miller-Rabin method determines that a given number is prime, given that after the computation of $n-1=2^sd$, $0{\leq}r{\leq}s-1$, the outcome satisfies $m^d{\equiv}1$(mod n) or $m^{2^rd}{\equiv}-1$(mod n). This paper proposes a step-by-step primality testing algorithm that restricts m=2, hence achieving 98.8% probability. The proposed method, as a first step, rejects composite numbers that do not satisfy the equation, $n=6k{\pm}1$, $n_1{\neq}5$. Next, it determines prime by computing $2^{2^{s-1}d}{\equiv}{\beta}_{s-1}$(mod n) and $2^d{\equiv}{\beta}_0$(mod n). In the third step, it tests ${\beta}_r{\equiv}-1$ in the range of $1{\leq}r{\leq}s-2$ for ${\beta}_0$ > 1. In the case of ${\beta}_0$ = 1, it retests m=3,5,7,11,13,17 sequentially. When applied to n=[101,1000], the proposed algorithm determined 96.55% of prime in the initial stage. The remaining 3% was performed for ${\beta}_0$ >1 and 0.55% for ${\beta}_0$ = 1.

• The Journal of the Korea institute of electronic communication sciences
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• v.12 no.6
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• pp.1181-1188
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• 2017

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.