Titlebook: Solving the Pell Equation; Michael J. Jacobson,Hugh C. Williams Textbook 20091st edition Springer-Verlag New York 2009 algebra.algebraic n

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The Ideal Class Group,lar, the regulator, defined in §5.3 to be the logarithm of the fundamental unit, is the number of bits of a fundamental solution up to a small constant factor. In Chapter 6 we have seen that the size of the regulator can vary greatly; indeed, there are many special families of discriminants that yie

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Some Additional Analytic Results,llary 8.35.1 for determining ., we notice that the expression for .(1, χΔ) is an infinite series. The purpose of this section is to derive a closed-form formula for .. To this end we must first consider some properties of a particular Gauss sum.

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The Subexponential Method,ation, have exponential complexity in the size of the discriminant .. The most exciting recent development has certainly been the discovery of a Las Vegas algorithm. by Buchmann for computing . and . whose expected running time is subexponential in log ¦.¦. This algorithm has enabled the computation

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Applications to Cryptography,f attacks on these systems are increasing rapidly, and where the impact of those attacks can be measured in billions of dollars and in the loss of reputation and personal integrity. Thus, it is essential today, more than ever, to develop techniques that will protect our communications. One essential

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