• Journal of Communications and Networks
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• v.15 no.1
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• pp.1-7
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• 2013

Most of the provably-secure proxy signature schemes rely on the average-case hardness problems such as the integer factorization problems and the discrete logarithm problems. Therefore, those schemes are insecure to quantum analysis algorithms, since there exist quantum algorithms efficiently solving the factorization and logarithm problems. To make secure proxy signature schemes against quantum analysis, some lattice-based proxy signature schemes are suggested. However, none of the suggested lattice-based proxy signature schemes is proxy-protected in the adaptive security model. In the paper, we propose a provably-secure ID-based proxy signature scheme based on the lattice problems. Our scheme is proxy-protected in the adaptive security model.

• Journal of the Korean Mathematical Society
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• v.45 no.6
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• pp.1613-1622
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• 2008

Let G be a graph with vertex set V(G) and edge set E(G), and let g, f be two nonnegative integer-valued functions defined on V(G) such that $g(x)\;{\leq}\;f(x)$ for every vertex x of V(G). We use $d_G(x)$ to denote the degree of a vertex x of G. A (g, f)-factor of G is a spanning subgraph F of G such that $g(x)\;{\leq}\;d_F(x)\;{\leq}\;f(x)$ for every vertex x of V(F). In particular, G is called a (g, f)-graph if G itself is a (g, f)-factor. A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let F = {$F_1$, $F_2$, ..., $F_m$} be a factorization of G and H be a subgraph of G with mr edges. If $F_i$, $1\;{\leq}\;i\;{\leq}\;m$, has exactly r edges in common with H, we say that F is r-orthogonal to H. If for any partition {$A_1$, $A_2$, ..., $A_m$} of E(H) with $|A_i|=r$ there is a (g, f)-factorization F = {$F_1$, $F_2$, ..., $F_m$} of G such that $A_i\;{\subseteq}E(F_i)$, $1\;{\leq}\;i\;{\leq}\;m$, then we say that G has (g, f)-factorizations randomly r-orthogonal to H. In this paper it is proved that every (0, mf - (m - 1)r)-graph has (0, f)-factorizations randomly r-orthogonal to any given subgraph with mr edges if $f(x)\;{\geq}\;3r\;-\;1$ for any $x\;{\in}\;V(G)$.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.12 no.5
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• pp.185-189
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• 2012

It is almost impossible to directly find the prime factor, p,q of a large semiprime, n=pq. So Most of the integer factorization algorithms uses a indirect method that find the prime factor of the p=GCD(a-b,n),q=GCD(a+b,n) after getting the congruence of squares of the $a^2{\equiv}b^2$(mod n). Many methods of getting the congruence of squares have proposed, but it is not easy to get with RSA number of greater than a 100-digit number. This paper proposes a fast algorithm to get the congruence of squares. The proposed algorithm succeeded in getting the congruence of squares to a 19-digit number.

• Journal of the Korea Society of Computer and Information
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• v.19 no.7
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• pp.71-76
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• 2014

There is to be virtually impossible to solve the very large digits of prime number p and q from composite number n=pq using integer factorization in typical public-key cryptosystems, RSA. When the public key e and the composite number n are known but the private key d remains unknown in an asymmetric-key RSA, message decryption is carried out by first obtaining ${\phi}(n)=(p-1)(q-1)=n+1-(p+q)$ and then using a reverse function of $d=e^{-1}(mod{\phi}(n))$. Integer factorization from n to p,q is most widely used to produce ${\phi}(n)$, which has been regarded as mathematically hard. Among various integer factorization methods, the most popularly used is the congruence of squares of $a^2{\equiv}b^2(mod\;n)$, a=(p+q)/2,b=(q-p)/2 which is more commonly used then n/p=q trial division. Despite the availability of a number of congruence of scares methods, however, many of the RSA numbers remain unfactorable. This paper thus proposes an algorithm that directly and immediately obtains ${\phi}(n)$. The proposed algorithm computes $2^k{\beta}_j{\equiv}2^i(mod\;n)$, $0{\leq}i{\leq}{\gamma}-1$, $k=1,2,{\ldots}$ or $2^k{\beta}_j=2{\beta}_j$ for $2^j{\equiv}{\beta}_j(mod\;n)$, $2^{{\gamma}-1}$ < n < $2^{\gamma}$, $j={\gamma}-1,{\gamma},{\gamma}+1$ to obtain the solution. It has been found to be capable of finding an arbitrarily located ${\phi}(n)$ in a range of $n-10{\lfloor}{\sqrt{n}}{\rfloor}$ < ${\phi}(n){\leq}n-2{\lfloor}{\sqrt{n}}{\rfloor}$ much more efficiently than conventional algorithms.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1525-1532
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• 2011

The $m{\times}n$ Boolean matrix A is said to be of Boolean rank r if there exist $m{\times}r$ Boolean matrix B and $r{\times}n$ Boolean matrix C such that A = BC and r is the smallest positive integer that such a factorization exists. We consider the the sets of matrix ordered pairs which satisfy extremal properties with respect to Boolean rank inequalities of matrices over nonbinary Boolean algebra. We characterize linear operators that preserve these sets of matrix ordered pairs as the form of $T(X)=PXP^T$ with some permutation matrix P.

• The KIPS Transactions:PartA
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• v.8A no.1
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• pp.36-41
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• 2001

In this paper, we first design an parallel quadratic sieve algorithm for factoring method. We then present parallel factoring algorithm for factoring a large odd integer by repeatedly using the parallel quadratic sieve algorithm based on the divide-and-conquer strategy on SIMD machines with DMM. We show that this algorithm is optimal in view of the product of time and processor numbers.

• Journal of Information Processing Systems
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• v.8 no.4
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• pp.567-574
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• 2012

This paper presents a strong designated verifiable signcryption scheme, in which a message is signcrypted by a signcryptor and only a specific receiver, who called a "designated verifier", verifies it using his own secret key. The scheme is secure, as an adversary can not verify the signature even if the secret key of the signer is compromised or leaked. The security of the proposed scheme lies in the complexity of solving two computationally hard problems, namely, the Discrete Logarithm Problem (DLP) and the Integer Factorization Problem (IFP). The security analysis of the scheme has been done and it is proved that, the proposed scheme can withstand an adaptive chosen ciphertext attack. This scheme can be very useful in organizations where there is a need to send confidential documents to a specific recipient. This scheme can also be applicable to real life scenarios, such as, e-commerce applications, e-banking and e-voting.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.16 no.1
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• pp.287-304
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• 2022

With the development of quantum computers, ring signature schemes based on large integer prime factorization, discrete logarithm problem, and bilinear pairing are under threat. For this reason, we design a ring signature scheme based on lattice with a fixed verification key. Compared with the previous ring signature scheme based on lattice, our design has a fixed verification key and does not disclose the signer's identity. Meanwhile, we propose an anonymous electronic voting scheme by using our ring signature scheme based on lattice and (t, n) threshold scheme, which makes up for the lack of current anonymous electronic voting that cannot resist attacks of the quantum computer. Finally, under standard model (SM), we prove that our ring signature scheme based on lattice is anonymous against the full-key exposure, and existentially non-forgeable against insider corruption. Furthermore, we also briefly analyze the security of our anonymous electronic voting scheme.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.17 no.1
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• pp.74-96
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• 2023

At present, the smart grid has become one of the indispensable infrastructures in people's lives. As a commonly used communication method, wireless communication is gradually, being widely used in smart grid systems due to its convenient deployment and wide range of serious challenges to security. For the insecurity of the schemes based on large integer factorization and discrete logarithm problem in the quantum environment, an identity-based key management scheme for smart grid over lattice is proposed. To assure the communication security, through constructing intra-cluster and inter-cluster multi-hop routing secure mechanism. The time parameter and identity information are introduced in the relying phase. Through using the symmetric cryptography algorithm to encrypt improve communication efficiency. Through output the authentication information with probability, the protocol makes the private key of the certification body no relation with the distribution of authentication information. Theoretic studies and figures show that the efficiency of keys can be authenticated, so the number of attacks, including masquerade, reply and message manipulation attacks can be resisted. The new scheme can not only increase the security, but also decrease the communication energy consumption.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.14 no.6
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• pp.25-31
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• 2014

When the public key e and the composite number n=pq are disclosed but not the private key d in an asymmetric-key RSA, message decryption is carried out by obtaining ${\phi}(n)=(p-1)(q-1)=n+1-(p+q)$ and subsequently computing $d=e^{-1}(mod{\phi}(n))$. The most commonly used decryption algorithm is integer factorization of n/p=q or $a^2{\equiv}b^2$(mod n), a=(p+q)/2, b=(q-p)/2. But many of the RSA numbers remain unfactorable. This paper therefore applies baby-step giant-step discrete logarithm and $2^k$-ary modular exponentiation to directly obtain ${\phi}(n)$. The proposed algorithm performs a reverse baby-step and $2^k$-ary adult-step. As a results, it reduces the execution time of basic adult-step to $1/2^k$ times and the memory $m={\lceil}\sqrt{n}{\rceil}$ to l, $a^l$ > n, hence obtaining ${\phi}(n)$ by executing within l times.