• Journal of the Korean Mathematical Society
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• v.52 no.4
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• pp.797-819
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• 2015

A trapdoor discrete logarithm group is a cryptographic primitive with many applications, and an algorithm that allows discrete logarithm problems to be solved faster using a pre-computed table increases the practicality of using this primitive. Currently, the distinguished point method and one extension to this algorithm are the only pre-computation aided discrete logarithm problem solving algorithms appearing in the related literature. This work investigates the possibility of adopting other pre-computation matrix structures that were originally designed for used with cryptanalytic time memory tradeoff algorithms to work as pre-computation aided discrete logarithm problem solving algorithms. We find that the classical Hellman matrix structure leads to an algorithm that has performance advantages over the two existing algorithms.

• Journal of KIISE:Computer Systems and Theory
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• v.29 no.11
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• pp.622-633
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• 2002

Authentication and Key Agreement using password provide convenience and amenity, but what human can remember has extremely low entropy. To overcome its defects, AMP(Authentiration and key agreement via Memorable Password) which performs authentication and key agreement securely via low entropy password are presented. AMP uses Diffie-Hellman problem that depends on discrete logarithm problem. Otherwise, this thesis applies elliptic curve cryptosystem to AMP for further efficiency That is, this thesis presents EC-AMP(Elliptic Curve-AMP) protocol based on elliptic curve discrete logarithm problem instead of discrete logarithm problem, and shows its high performance through the implementation. EC-AMP secures against various attacks in the random oracle model just as AMP Thus, we nay supply EC-AMP to the network environment that requires authentication and key agreement to get both convenience and security from elliptic curve discrete logarithm problem.

• Journal of Communications and Networks
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• v.4 no.2
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• pp.81-89
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• 2002

In this paper, we design and implement an efficient fair off-line electronic cash system based on Elliptic Curve Discrete Logarithm Problem (ECDLP), in which the anonymity of coins is revocable by a trustee in case of dispute. To achieve this, we employ the Petersen and Poupard s electronic cash system [1] and extend it by using an elliptic curve over the finite field GF($2^n$). This naturally reduces message size by 85% compared with the original scheme and makes a smart card to store coins easily. Furthermore, we use the Baek et al. s provably secure public key encryption scheme [2] to improve the security of electronic cash system. As an extension, we propose a method to add atomicity into new electronic cash system. To the best of our knowledge, this is the first result to implement a fair off-line electronic cash system based on ECDLP with provable security.

• Proceedings of the KIEE Conference
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• 2001.07d
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• pp.2576-2578
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• 2001

Elliptic curve cryptosystems based on discrete logarithm problem in the group of points of an elliptic curve defined over a finite field. The discrete logarithm in an elliptic curve group appears to be more difficult than discrete logarithm problem in other groups while using the relatively small key size. An implementation of elliptic curve cryptosystems needs finite field arithmetic computation. Hence finite field arithmetic modules must require less hardware resources to archive high performance computation. In this paper, a new architecture of finite field multiplier using conversion scheme of normal basis representation into polynomial basis representation is discussed. Proposed architecture provides less resources and lower complexity than conventional bit serial multiplier using normal basis representation. This architecture has synthesized using synopsys FPGA express successfully.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.13 no.2
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• pp.91-98
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• 2003

We introduce a new public key system based on the discrete logarithm Problem(DLP) in a quotient group of finite fields. This system achieves savings not only in communication overhead by reducing public key size and transfer data by half but also in computational costs by performing efficient exponentiation. In particular, this system takes about 50% speed-up, compared to DSA which has the same security.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.1 no.1
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• pp.97-101
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• 1991

The polynomial factorization problem is important not only number theorly but chyptology with Discrete logarithm. We factorized polynolmials with integer coefficients by means of factori-zing polynomials on a finite field by Hensel's Lifting Lemma and finding factors of pol;ynomial with integer coeffcients.

• Journal of Information Processing Systems
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• v.6 no.3
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• pp.335-346
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• 2010

The present study investigates the difficulty of solving the mathematical problem, namely the DLP (Discrete Logarithm Problem) for ephemeral keys. The DLP is the basis for many public key cryptosystems. The ephemeral keys are used in such systems to ensure security. The DLP defined on a prime field $Z^*_p of random prime is considered in the present study. The most effective method to solve the DLP is the ICM (Index Calculus Method). In the present study, an efficient way of computing the DLP for ephemeral keys by using a new variant of the ICM when the factors of p-1 are known and small is proposed. The ICM has two steps, a pre-computation and an individual logarithm computation. The pre-computation step is to compute the logarithms of a subset of a group and the individual logarithm step is to find the DLP using the precomputed logarithms. Since the ephemeral keys are dynamic and change for every session, once the logarithms of a subset of a group are known, the DLP for the ephemeral key can be obtained using the individual logarithm step. Therefore, an efficient way of solving the individual logarithm step based on the newly proposed precomputation method is presented and the performance is analyzed using a comprehensive set of experiments. The ephemeral keys are also solved by using other methods, which are efficient on random primes, such as the Pohlig-Hellman method, the Van Oorschot method and the traditional individual logarithm step. The results are compared with the newly proposed individual logarithm step of the ICM. Also, the DLP of ephemeral keys used in a popular password key exchange protocol known as Chang and Chang are computed and reported to launch key recovery attack.

• Journal of Multimedia Information System
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• v.8 no.1
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• pp.45-56
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• 2021

This document traces the new technologies development based on a deep classical Hill method improvement. Based on the chaos, this improvement begins with the 256 element body construction, which is to replace the classic ring used by all encryption systems. In order to facilitate the application of algebraic operators on the pixels, two substitution tables will be created, the first represents the discrete logarithm, while the second represents the discrete exponential. At the same time, a large invertible matrix whose structure will be explained in detail will be the subject of the advanced classical Hill technique improvement. To eliminate any linearity, this matrix will be accompanied by dynamic vectors to install an affine transformation. The simulation of a large number of images of different sizes and formats checked by our algorithm ensures the robustness of our method.

• Journal of the Korea Society of Computer and Information
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• v.20 no.8
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• pp.29-33
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• 2015

This paper proposes a discrete logarithm algorithm that remarkably reduces the execution time of Pollard's Rho algorithm. Pollard's Rho algorithm computes congruence or collision of ${\alpha}^a{\beta}^b{\equiv}{\alpha}^A{\beta}^B$ (modp) from the initial value a = b = 0, only to derive ${\gamma}$ from $(a+b{\gamma})=(A+B{\gamma})$, ${\gamma}(B-b)=(a-A)$. The basic Pollard's Rho algorithm computes $x_i=(x_{i-1})^2,{\alpha}x_{i-1},{\beta}x_{i-1}$ given ${\alpha}^a{\beta}^b{\equiv}x$(modp), and the general algorithm computes $x_i=(x_{i-1})^2$, $Mx_{i-1}$, $Nx_{i-1}$ for randomly selected $M={\alpha}^m$, $N={\beta}^n$. This paper proposes 4-model Pollard Rho algorithm that seeks ${\beta}_{\gamma}={\alpha}^{\gamma},{\beta}_{\gamma}={\alpha}^{(p-1)/2+{\gamma}}$, and ${\beta}_{{\gamma}^{-1}}={\alpha}^{(p-1)-{\gamma}}$) from $m=n={\lceil}{\sqrt{n}{\rceil}$, (a,b) = (0,0), (1,1). The proposed algorithm has proven to improve the performance of the (0,0)-basic Pollard's Rho algorithm by 71.70%.

• Journal of the Korea Institute of Information Security & Cryptology
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• v.9 no.3
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• pp.3-12
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• 1999

The public key crytptosystem is represented by RSA based on the difficulty of integer factorization and ElGamal cryptosystem based on the intractability of the discrete logarithm problem in a cyclic group G. The index-calculus algorithm for discrete logarithms in GF${$q^n$}^+$ requires an polynomial factorization. The Niederreiter recently developed deterministic facorization algorithm for polynomial over GF$q^n$ In this paper we implemented the arithmetic of finite field with c-language and gibe an implementation of the Niederreiter's algorithm over GF$2^n$ using normal bases.