• Communications of the Korean Mathematical Society
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• v.20 no.4
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• pp.645-648
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• 2005

Let K be a Galois extension of a field F with G = Gal(K/F). Let L be an extension of F such that $K\;{\otimes}_F\;L\;=\; N_1\;{\oplus}N_2\;{\oplus}{\cdots}{\oplus}N_k$ with corresponding primitive idempotents $e_1,\;e_2,{\cdots},e_k$, where Ni's are fields. Then G acts on $\{e_1,\;e_2,{\cdots},e_k\}$ transitively and $Gal(N_1/K)\;{\cong}\;\{\sigma\;{\in}\;G\;/\;{\sigma}(e_1)\;=\;e_1\}$. And, let R be a commutative F-algebra, and let P be a prime ideal of R. Let T = $K\;{\otimes}_F\;R$, and suppose there are only finitely many prime ideals $Q_1,\;Q_2,{\cdots},Q_k$ of T with $Q_i\;{\cap}\;R\;=\;P$. Then G acts transitively on $\{Q_1,\;Q_2,{\cdots},Q_k\},\;and\;Gal(qf(T/Q_1)/qf(R/P))\;{\cong}\;\{\sigma{\in}\;G/\;{\sigma}-(Q_1)\;=\;Q_1\}$ where qf($T/Q_1$) is the quotient field of $T/Q_1$.

• Journal of Advanced Navigation Technology
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• v.16 no.1
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• pp.62-69
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• 2012

Multiple-control Toffoli(MCT) gates are macro-level multiple-valued gates needing quantum technology dependent primitive gates, and have been used in Galois Field sum-of-product (GFSOP) based synthesis of quantum logic circuit. Reversible logic is very important in quantum computing for low-power circuit design. This paper presents a reversible GF4 multiplier at first, and GF4 multiplier based quaternary MCT gate realization is also proposed. In the comparisons of MCT gate realization, we show the proposed MCT gate can reduce considerably primitive gates and delays in contrast to the composite one of the smaller MCT gates in proportion to the multiple-control input increase.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.15 no.9
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• pp.3458-3481
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• 2021

Existing methods for blind identification of linear block codes without a candidate set are mainly built on the Gauss elimination process. However, the fault tolerance will fall short when the intercepted bit error rate (BER) is too high. To address this issue, we apply the reverse algebra approach and propose a novel "two-step-screening" algorithm by solving the linear error equations on the binary Galois field, or GF(2). In the first step, a recursive matrix partition is implemented to solve the system linear error equations where the coefficient matrix is constructed by the full codewords which come from the intercepted noisy bitstream. This process is repeated to derive all those possible parity-checks. In the second step, a check matrix constructed by the intercepted codewords is applied to find the correct parity-checks out of all possible parity-checks solutions. This novel "two-step-screening" algorithm can be used in different codes like Hamming codes, BCH codes, LDPC codes, and quasi-cyclic LDPC codes. The simulation results have shown that it can highly improve the fault tolerance ability compared to the existing Gauss elimination process-based algorithms.

• KSII Transactions on Internet and Information Systems (TIIS)
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• v.13 no.9
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• pp.4788-4813
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• 2019

In this paper, we introduce concepts of optimal and near optimal secret data hiding schemes. We present a new digital image steganography approach based on the Galois field $GF(p^m)$ using graph and automata to design the data hiding scheme of the general form ($k,N,{\lfloor}{\log}_2p^{mn}{\rfloor}$) for binary, gray and palette images with the given assumptions, where k, m, n, N are positive integers and p is prime, show the sufficient conditions for the existence and prove the existence of some optimal and near optimal secret data hiding schemes. These results are derived from the concept of the maximal secret data ratio of embedded bits, the module approach and the fastest optimal parity assignment method proposed by Huy et al. in 2011 and 2013. An application of the schemes to the process of hiding a finite sequence of secret data in an image is also considered. Security analyses and experimental results confirm that our approach can create steganographic schemes which achieve high efficiency in embedding capacity, visual quality, speed as well as security, which are key properties of steganography.

• Journal of the Chungcheong Mathematical Society
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• v.27 no.2
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• pp.205-210
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• 2014

The author with C. H. Kim and Y. Lee constructed infinite families of elliptic curves over cubic number fields K with prescribed torsion groups which occur infinitely often. In this paper, we examine the Galois structures of such cubic number fields K for the families of elliptic curves with cyclic torsion.

• Honam Mathematical Journal
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• v.38 no.1
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• pp.85-93
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• 2016

Let A be an abelian variety defined over a number field K and let L be a degree 3 non-Galois extension of K. Let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups of A over K and over L. Assuming that III(A/L) is finite, we compute [III(A/K)][III($A_{\varphi}/K$)]/[III(A/L)], where [X] is the order of a finite abelian group X.

• Korean Journal of Mathematics
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• v.26 no.3
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• pp.537-544
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• 2018

There is a standard construction of lifting cyclic codes over the prime finite field ${\mathbb{Z}}_p$ to the rings ${\mathbb{Z}}_{p^e}$ and to the ring of p-adic integers. We generalize this construction for arbitrary finite fields. This will naturally enable us to lift codes over finite fields ${\mathbb{F}}_{p^r}$ to codes over Galois rings GR($p^e$, r). We give concrete examples with all of the lifts.

• Journal of the Chungcheong Mathematical Society
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• v.21 no.1
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• pp.61-69
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• 2008

For a real subextension of some cyclotomic function field with a non-cyclic Galois group order $l^2$, l being a prime different from the characteristic of function field, we compute the index of the Sinnott group of circular units.

• Journal of the Korean Institute of Telematics and Electronics
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• v.23 no.4
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• pp.460-465
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• 1986

In this paper, a method for constructing of the sequential multiple-valued logic circuits over Galois field GF(px) is proposed. First, we derive the Talyor series over Galois field and the unique matrices which accords with the number of the element over the finite field, and we constdruct sequential multiple-valued logic circuits using these matrices. Computational procedure for traditional polynomial expansion can be reduced by using this method. Also, single and multi-input circuits can be easily implemented.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.83-96
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• 2013

Let K be a number field and fix a prime number $p$. For any set S of primes of K, we here say that an elliptic curve E over K has S-reduction if E has bad reduction only at the primes of S. There exists the set $B_{K,p}$ of primes of K satisfying that any elliptic curve over K with $B_{K,p}$-reduction has no $p$-torsion points under certain conditions. The first aim of this paper is to construct elliptic curves over K with $B_{K,p}$-reduction and a $p$-torsion point. The action of the absolute Galois group on the $p$-torsion subgroup of E gives its associated Galois representation $\bar{\rho}_{E,p}$ modulo $p$. We also study the irreducibility and surjectivity of $\bar{\rho}_{E,p}$ for semistable elliptic curves with $B_{K,p}$-reduction.