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Algebraic formulation and program generation of threedimensional hilbert spacefilling curves
 In The 2004 International Conference on Imaging Science, Systems, and Technology
, 2004
"... Abstract: We use a tensor product based multilinear algebra theory to formulate threedimensional Hilbert spacefilling curves. A 3D Hilbert spacefilling curve is specified as a permutation which rearranges threedimensional 2 n 2 n 2 n data elements stored in the row major order as in C language ..."
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Abstract: We use a tensor product based multilinear algebra theory to formulate threedimensional Hilbert spacefilling curves. A 3D Hilbert spacefilling curve is specified as a permutation which rearranges threedimensional 2 n 2 n 2 n data elements stored in the row major order as in C language or the column major order as in FORTRAN language to the order of traversing a 3D Hilbert spacefilling curve. The tensor product formulation of 3D Hilbert spacefilling curves uses stride permutation, reverse permutation, and Gray permutation. We present both recursive and iterative tensor product formulas of 3D Hilbert spacefilling curves. In addition, we derive a tensor product formula of inverse 3D Hilbert spacefilling curve permutation. The tensor product formulas are directly translated into computer programs which can be used in various applications. The process of program generation is explained in the paper.
VLSI circuit design of matrix transposition using tensor product formulation
 Dept. of Computer Science and Information Engineering, National Dong Hwa University
, 2003
"... Matrix transposition is a simple, but an important computational problem. It explores many key issues on data locality. In this paper, we will design matrix transposition algorithms on various interconnection networks for VLSI circuit design, including omega, baseline and hypercube networks. Since d ..."
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Matrix transposition is a simple, but an important computational problem. It explores many key issues on data locality. In this paper, we will design matrix transposition algorithms on various interconnection networks for VLSI circuit design, including omega, baseline and hypercube networks. Since different interconnection networks have their own architectural characteristics and properties, an algorithm needs to be tuned in order to be efficiently implemented on various networks. We use a tensor product based algebraic theory to design matrix transposition algorithms on various interconnection networks. After designing matrix transposition on various interconnection networks, we use a hardware description language, Verilog, to realize algorithms on FPGA. A major goal of this paper is to provide an effective way for designing VLSI circuits of DSP algorithms.
A Programming Methodology for Designing Block Recursive Algorithms
 Technol., Month 200X, Vol.21, No.X
"... In this paper, we use the tensor product notation as the framework of a programming methodology for designing block recursive algorithms. We first express a computational problem in its matrix form. Next, we formulate a matrix equation for the matrix of the computational problem. Then, we try to fin ..."
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In this paper, we use the tensor product notation as the framework of a programming methodology for designing block recursive algorithms. We first express a computational problem in its matrix form. Next, we formulate a matrix equation for the matrix of the computational problem. Then, we try to find a solution of the matrix equation such that the solution is composed of simple matrices. Finally, we recursively factorize the subproblem to obtain a tensor product formula representing an algorithm for the given problem. In this methodology, the operations of a tensor product formula can be mapped to language constructs of highlevel programming languages. That is, we can generate computer programs, including programs for parallel computers and distributedmemory multiprocessors, from tensor product formulas. In this paper, we use the parallel prefix problem and the discrete Fourier transform problem as examples to illustrate the methodology and derive various parallel prefix and fast Fourier transform algorithms.
IMPLEMENTATION OF KARATSUBA ALGORITHM USING POLYNOMIAL MULTIPLICATION
"... Efficiency in multiplication is very important in applications like signal processing, cryptosystems and coding theory. This paper presents the design of a fast multiplier using the Karatsuba algorithm to multiply two numbers using the technique of polynomial multiplication. The Karatsuba algorithm ..."
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Efficiency in multiplication is very important in applications like signal processing, cryptosystems and coding theory. This paper presents the design of a fast multiplier using the Karatsuba algorithm to multiply two numbers using the technique of polynomial multiplication. The Karatsuba algorithm saves coefficient multiplications at the cost of extra additions as compared to the ordinary multiplication method. The Karatsuba algorithm is more efficient for multiplication of large numbers.
VLSI Circuit Design of Digital Signal Processing Algorithms Using Tensor Product Formulation
"... Abstract Many important computation problems can be specified by block recursive algorithms. For example, matrix transposition and fast Fourier transform are block recursive algorithms. In this paper, we present a methodology of VLSI circuit design for block recursive algorithms based on the tensor ..."
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Abstract Many important computation problems can be specified by block recursive algorithms. For example, matrix transposition and fast Fourier transform are block recursive algorithms. In this paper, we present a methodology of VLSI circuit design for block recursive algorithms based on the tensor product theory. Matrix transposition and fast Fourier transform algorithms are designed and implemented following this methodology. First, matrix transposition and fast Fourier transform algorithms are expressed as tensor product formulas. The tensor product formulas are modified to fit into interconnection networks, including the omega network and the hypercube network. The formulas are then used to generate highlevel programming language code. Finally, a hardware description language, Verilog, is used to realize the algorithms according to the generated programs. The major goal of this paper is to provide an effective way to design VLSI circuits for block recursive algorithms.
Ultra Long Integer Multiplication on GDPS
"... Many Internet applications require intensive cryptographic calculation such as publickey encryptions and digital signatures. These schemes require a computation of large integer multiplications. Those cryptographic schemes are vulnerable to a bruteforce attack, and the large key is the countermeas ..."
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Many Internet applications require intensive cryptographic calculation such as publickey encryptions and digital signatures. These schemes require a computation of large integer multiplications. Those cryptographic schemes are vulnerable to a bruteforce attack, and the large key is the countermeasure. In practice, the key size that makes bruteforce attack impractical will slows down the speed of encryption and decryption. Multiplication of two very long integers usually takes time to compute. Distributed Karatsuba algorithm is proposed to reduce the time of multiplication of two very long digits. The proposed architecture that makes use of Karatsuba algorithm achieves faster multiplication.