Elliptic curves number theory

"Ebook Elliptic curves number theory and cryptography (Second edition)" present Taking a basic approach to elliptic curves, this accessible book prepares readers to tackle more advanced problems in the field. It introduces elliptic curves over finite fields early in the text, before moving on to interesting applications, such as cryptography, factoring, and primality testing. The book also discusses the use of elliptic curves in Fermat’s Last Theorem. Relevant abstract algebra material on group theory and fields can be found in the appendices.

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Document present cryptosystems based on elementary number theory, cryptosystems based on elliptic curves, elementary number theory background, diffie-hellman key exchange, elgamal protocol, rsa cryptosystem...

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Locally symmetric spaces play an important part in differential geometry and arise from many different areas such as topology, number theory, representation theory, algebraic geometry,...The typical important class consists of quotients of symmetric spaces by arithmetic groups, for example, the moduli space of elliptic curves is the quotient of the upper half plane H2 by SL(2, Z).

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We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let K− denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the generator c of Gal(K/k) acts on Gal(K− /K) by inversion). We prove (under mild hypotheses on p) that if the Zp -rank of the pro-p Selmer group Sp...

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The discovery of infinite products byWallis and infinite series by Newton marked the beginning of the modern mathematical era. The use of series allowed Newton to find the area under a curve defined by any algebraic equation, an achievement completely beyond the earlier methods ofTorricelli, Fermat, and Pascal. The work of Newton and his contemporaries, including Leibniz and the Bernoullis, was concentrated in mathematical analysis and physics.

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