A course in number theory and cryptography
The purpose of this book is to introduce the reader to arithmetic topics, both ancient and modern, that have been at the center of interest in applications of number theory, particularly in cryptography. No background in algebra or number theory is assumed, and the book begins with a discussion of the basic number theory that is needed. The approach taken is algorithmic, emphasizing estimates of the efficiency of the techniques that arise from the theory. A special feature is the inclusion of recent application of the theory of elliptic curves. Extensive exercises and careful answers have been included in all of the chapters. Because number theory and cryptography are fast-moving fields, this new edition contains substantial revisions and updated references.
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Finite Fields and Quadratic Residues
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1-to-l correspondence 26-letter alphabet b2 mod base binary bit operations required Carmichael number Chinese Remainder Theorem choose ciphertext coefficients composite compute congruence continued fraction Corollary cryptography deciphering key deciphering matrix denote digits digraphs discrete log problem divide divisible elliptic curve enciphering key enciphering transformation encryption equation estimate Euclidean algorithm Euler pseudoprime example Exercise Fermat's Little Theorem field F finite field follows formula Gaussian integers gives inverse irreducible polynomials JV-letter knapsack problem large prime least nonnegative residue Legendre symbol letters monic irreducible polynomials multiple nonresidue number of bit number theory numerical equivalents obtain pair plaintext message units positive integer possible primality test prime factors prime number prime power Prove public key cryptosystem quadratic real number residue modulo rho method root of unity satisfies solution solve the discrete square root strong pseudoprime suppose vector write Z/nZ zero