Titlebook: Number Theoretic Methods in Cryptography; Complexity lower bou Igor Shparlinski Book 1999 Springer Basel AG 1999 complexity.complexity theo

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Approximation of the Discrete Logarithm Modulo Here we show that polynomials and algebraic functions approximating the discrete logarithm modulo . on sufficiently large sets must be of sufficiently large degree, in fact exponentially large (in terms of log.). Many of the results of this chapter can also be found in [48]. We start with a rather simple statement.

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Approximation of the Discrete Logarithm Modulo , - 1In this chapter we consider various approximations and representations of the discrete logarithm modulo a divisor . of . - 1. Certainly the case of . = 2 is of special interest because it corresponds to representation of the rightmost bit of ind ..

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Approximation of the Discrete Logarithm by Boolean FunctionsHere we consider the bitwise approximation of the discrete logarithm given the bit representation of the argument. Moreover, we concentrate on the rightmost bit of ind .. This question is essentially equivalent to deciding quadratic residuacity of ..

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Approximation of the Discrete Logarithm by Real and Complex PolynomialsHere we consider some questions about approximation of the discrete logarithm by real and even complex polynomials. Unfortunately our results are weaker that those in our previous settings.

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Special Polynomials and Boolean FunctionsIn this chapter we show how to apply the techniques of this book to various questions about permutation polynomials, powers ., Zech’s logarithm, primitive root testing and symmetric Boolean functions.

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RSA and Blum—Blum—Shub Generators of Pseudo-Random NumbersLet ϑ, . and . be integers such that gcd(ϑ, .) = 1.

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Further DirectionsWe now try very briefly to describe several other problems which look quite unrelated to the questions considered in this book (in particular, they are not directly related to any functions over a finite field), but for which, nevertheless, we hope our approach could turn out to be useful.

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978-3-0348-9723-5Springer Basel AG 1999

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