By Oded Goldreich
ISBN-10: 354064766X
ISBN-13: 9783540647669
Cryptography is without doubt one of the such a lot lively parts in present arithmetic examine and purposes. This publication makes a speciality of cryptography in addition to comparable parts: the examine of probabilistic facts structures, and the idea of computational pseudorandomness. Following a standard subject that explores the interaction among randomness and computation, the real notions in every one box are lined, in addition to novel rules and insights.
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Extra info for Modern Cryptography, Probabilistic Proofs and Pseudorandomness (Algorithms and Combinatorics)
Example text
An e−n/x = O x 1+ . n=1 Thus, for any individual term in the sum, we have an e−n/x = O x 1+ . Choosing x = n, we deduce that an = O(n1+ ). If we let π be an automorphic representation on GL(2) with local parameters, αp , βp , then one can give an intuitive description of the symmetric power Lfunctions attached to π in the following way. Consider, with rm denoting the m-th symmetric power of π , m m−j 1− Lm (s) := L(s, π, rm ) = p j =1 αp j βp −1 ps where we are ignoring the finitely many Euler factors that need to be modified corresponding to the ramified factors.
The subspace S is invariant under both of these actions. Diagonalizing the Hecke action, we get a decomposition S = ⊕f S (f ) into submodules, where each S (f ) is a rank two module over Kf [Gal(Q/Q)], and Kf is a certain number field (the field generated by the Fourier coefficients of f ). 5 Geometric Realization of Modular Forms of Higher Weight 37 These are the Galois representations attached to modular forms of higher weight whose existence had been conjectured by Serre and constructed by Deligne [38].
1007/978-81-322-0770-2_4, © Springer India 2013 39 40 4 The Ramanujan Conjecture from GL(2) to GL(n) wait until 1974 when Deligne proved it as a consequence of his proof of the Weil conjectures. We outline the proofs of these conjectures. As mentioned in the introduction to Chap. 3, the essential property of (z) is that it is a modular form of weight 12 for the full modular group SL2 (Z), which is the group of 2 × 2 matrices with integer entries and determinant 1. This means that for z in the upper half-plane h, we have az + b cz + d = (cz + d)12 (z) a c ∀ b ∈ SL2 (Z).
Modern Cryptography, Probabilistic Proofs and Pseudorandomness (Algorithms and Combinatorics) by Oded Goldreich
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