• 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 Korea Institute of Information and Communication Engineering
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• v.9 no.8
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• pp.1774-1779
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• 2005

This paper propose the design method of the constructing the digital logic systems over galois fields using by the galois field decision diagram(GFDD) that is based on the graph theory. The proposed design method is as following. First of all, we discuss the mathematical properties of the galois fields and the basic properties of the graph theory. After we discuss the operational domain and the functional domain, we obtain the transformation matrixes, $\psi$GF(P)(1) and $\xi$GF(P)(1), in the case of one variable, that easily manipulate the relationship between two domains. And we extend above transformation matrixes to n-variable case, we obtain $\psi$GF(P)(1) and $\xi$GF(P)(1). We discuss the Reed-Muller expansion in order to obtain the digital switching functions of the P-valued single variable. And for the purpose of the extend above Reed-Muller expansion to more two variables, we describe the Kronecker product arithmetic operation.

• East Asian mathematical journal
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• v.38 no.5
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• pp.583-591
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• 2022

Let K be an imaginary quadratic field and 𝒪 be an order in K. Let H_𝒪 be the ring class field of 𝒪. Furthermore, for a positive integer N, let K_𝒪,N be the ray class field modulo N𝒪 of 𝒪. When the discriminant of 𝒪 is different from -3 and -4, we construct an extended form class group which is isomorphic to the Galois group Gal(K_𝒪,N/H_𝒪) and describe its Galois action on K_𝒪,N in a concrete way.

• Kyungpook Mathematical Journal
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• v.56 no.3
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• pp.755-761
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• 2016

Let A be an abelian variety over a global field K. We show that, in "many" cases, Chen-Kuan's invariant M(A[n]), that is the average number of n-torsion points of A over various residue fields of K, has the minimal possible value.

• Proceedings of the Korean Institute of Information and Commucation Sciences Conference
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• v.9 no.1
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• pp.880-883
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• 2005

This paper presents a method of constructing the sequential logic machines over finite fields(or galois fields). The proposed the sequential logic machines is constructed by as following. First of all, we obtain the linear characteristics between present state and next state based on mathematical properties of finite fields and sequential logic machines. Next, we realize the sequential logic machines over finite field GF(P) using above linear characteristics and characteristic polynomial that expressed using by matrix.

• Journal of the Korean Mathematical Society
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• v.22 no.2
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• pp.117-124
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• 1985

• Journal of the Institute of Electronics Engineers of Korea SC
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• v.41 no.6
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• pp.27-35
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• 2004

Finite or Galois fields have been used in numerous applications such as error correcting codes, digital signal processing and cryptography. These applications often require exponetiation on GF(2$^{m}$ ) which is a very computationally intensive operation. Most of the existing methods implemented the exponetiation by iterative methods using repeated multiplications, which leads to much computational load, or needed much hardware cost because of their structural complexity in implementing. In this paper, we present an effective VLSI architecture for exponentiation on GF(2$^{m}$ ). This circuit computes the exponentiation by multiplying product terms, each of which corresponds to an exponent bit. Until now use of this type algorithm has been confined to a primitive element but we generalize it to any elements in GF(2$^{m}$ ).

• Journal of the Korean Mathematical Society
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• v.39 no.2
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• pp.289-317
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• 2002

We associate binary codes to polynomials over fields of characteristic two and show that the binary Golay codes are associated to the Mathieu group polynomials in characteristics two. We give two more polynomials whose Galois group in $M_{12}$ but different self-orthogonal binary codes are associated. Also, we find a family of $M_{24}$-coverings which includes previous ones.

• 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 Korea Society of Digital Industry and Information Management
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• v.18 no.1
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• pp.1-9
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• 2022

Galois field arithmetic is important in error correcting codes and public-key cryptography schemes. Hardware realization of these schemes requires an efficient implementation of Galois field arithmetic operations. Multiplication is the main finite field operation and designing efficient multiplier can clearly affect the performance of compute-intensive applications. Diverse algorithms and hardware architectures are presented in the literature for hardware realization of Galois field multiplication to acquire a reduction in time and area. This paper presents a low complexity semi-systolic multiplier to facilitate parallel processing by partitioning Montgomery modular multiplication (MMM) into two independent and identical units and two-level systolic computation scheme. Analytical results indicate that the proposed multiplier achieves lower area-time (AT) complexity compared to related multipliers. Moreover, the proposed method has regularity, concurrency, and modularity, and thus is well suited for VLSI implementation. It can be applied as a core circuit for multiplication and division/exponentiation.