Publication 2009 Issue No. 2 - February Abstract - A Discrete Logarithm Number System for Integer Arithmetic Modulo 2^{k}: Algorithms and Lookup Structures
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A Discrete Logarithm Number System for Integer Arithmetic Modulo 2^{k}: Algorithms and Lookup Structures
February 2009 (vol. 58 no. 2)
pp. 163-174
 ASCII Text x Alexandru Fit-Florea, Lun Li, Mitchell A. Thornton, David W. Matula, "A Discrete Logarithm Number System for Integer Arithmetic Modulo 2^{k}: Algorithms and Lookup Structures," IEEE Transactions on Computers, vol. 58, no. 2, pp. 163-174, February, 2009.
 BibTex x @article{ 10.1109/TC.2008.204,author = {Alexandru Fit-Florea and Lun Li and Mitchell A. Thornton and David W. Matula},title = {A Discrete Logarithm Number System for Integer Arithmetic Modulo 2^{k}: Algorithms and Lookup Structures},journal ={IEEE Transactions on Computers},volume = {58},number = {2},issn = {0018-9340},year = {2009},pages = {163-174},doi = {http://doi.ieeecomputersociety.org/10.1109/TC.2008.204},publisher = {IEEE Computer Society},address = {Los Alamitos, CA, USA},}
 RefWorks Procite/RefMan/Endnote x TY - JOURJO - IEEE Transactions on ComputersTI - A Discrete Logarithm Number System for Integer Arithmetic Modulo 2^{k}: Algorithms and Lookup StructuresIS - 2SN - 0018-9340SP163EP174EPD - 163-174A1 - Alexandru Fit-Florea, A1 - Lun Li, A1 - Mitchell A. Thornton, A1 - David W. Matula, PY - 2009KW - High-speed arithmeticKW - arithmetic and logic unitsKW - computer arithmeticKW - discrete logarithmKW - number encodingsKW - conversionsKW - bit serialKW - integer powerKW - table lookup.VL - 58JA - IEEE Transactions on ComputersER -
Alexandru Fit-Florea, Advanced Micro Devices, Inc., Sunnyvale
Lun Li, Texas Instruments, Dallas
Mitchell A. Thornton, Southern Methodist University, Dallas
David W. Matula, Southern Methodist University, Dallas
We present a k-bit encoding of the k-bit binary integers based on a discrete logarithm representation. The representation supports a discrete logarithm number system (DLS) that allows integer multiplication to be reduced to addition and integer exponentiation to be reduced to multiplication. We introduce right-to-left bit serial conversion, deconversion, and unified conversion/deconversion algorithms between binary and DLS. The conversion algorithms utilize {\rm O}(k) additions, do not require the use of a multiplier, and are applicable at least up to 128-bit integers. We illustrate the use of the representation in determining a novel and efficient integer power modulo 2^{k} operation |x^{y}|_{2^{k}} and compare hardware performance with a current state-of-the-art method. Furthermore, we describe properties of the conversion mappings that allow compact table lookup structures to be employed for direct conversion to and deconversion from the DLS encoding. Our lookup architecture allows 16-bit conversion and deconversion mappings to be realized with table sizes of order 2-8 Kbytes, which is up to a 64\times size reduction of the 128 Kbytes of an arbitrary 16-bits-in, 16-bits-out function table. Performance and area results that demonstrate the effectiveness of the table lookup architecture are given. The lookup methodology extends to other 16-bit integer functions such as multiplicative inverse and squaring operations.

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Index Terms:
High-speed arithmetic, arithmetic and logic units, computer arithmetic, discrete logarithm, number encodings, conversions, bit serial, integer power, table lookup.
Citation:
Alexandru Fit-Florea, Lun Li, Mitchell A. Thornton, David W. Matula, "A Discrete Logarithm Number System for Integer Arithmetic Modulo 2^{k}: Algorithms and Lookup Structures," IEEE Transactions on Computers, vol. 58, no. 2, pp. 163-174, Feb. 2009, doi:10.1109/TC.2008.204