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Full-Text Articles in Physical Sciences and Mathematics
Solvability Characterizations Of Pell Like Equations, Jason Smith
Solvability Characterizations Of Pell Like Equations, Jason Smith
Boise State University Theses and Dissertations
Pell's equation has intrigued mathematicians for centuries. First stated as Archimedes' Cattle Problem, Pell's equation, in its most general form, X2 – P • Y2 = 1, where P is any square free positive integer and solutions are pairs of integers, has seen many approaches but few general solutions. The eleventh century Indian mathematician Bhaskara solved X2 – 61 • Y2 = 1 and, in response to Fermat's challenge, Wallis and Brouncker gave solutions to X2 – 151 • Y2 = 1 and X2 –313 • Y2 = 1. Fermat claimed to …
Statistical Investigation Of Structure In The Discrete Logarithm, Andrew Hoffman
Statistical Investigation Of Structure In The Discrete Logarithm, Andrew Hoffman
Mathematical Sciences Technical Reports (MSTR)
The absence of an efficient algorithm to solve the Discrete Logarithm Problem is often exploited in cryptography. While exponentiation with a modulus is extremely fast with a modern computer, the inverse is decidedly not. At the present time, the best algorithms assume that the inverse mapping is completely random. Yet there is at least some structure, and to uncover additional structure that may be useful in constructing or refining algorithms, statistical methods are employed to compare modular exponential mappings to random mappings. More concretely, structure will be defined by representing the mappings as functional graphs and using parameters from graph …