Elliptic curves : number theory and cryptography /

Diophantine analysis, solution by integers of polynomial equations, provably turns intractable as the degree and number of variables grow large. For now, elliptic curves (two variable cubic equations) lie at the threshold of tractability: celebrated unproved conjectures, especially that of Birch and SwinnertonDyer, promise an algorithm for finding all the rational points on an elliptic curve. Elliptic curves play a key role in Wiles's solution of Fermat's Last Theorem and find many applications in cryptography and factorization algorithms. Washington (Univ. of Maryland, College Park) has found just the right level of abstraction for a first book; he gets very far and he demands less algebraic geometry than J. Silverman uses in his two books and also less analysis than A. Knapp presumes in his. Notably, he offers the most lucid and concrete account ever of the perpetually mysterious ShafarevichTate group. A pleasure to read! ^BSumming Up: Highly recommended. Upper-division undergraduates through faculty. D. V. Feldman University of New Hampshire