An Introduction to Mathematical Cryptography
ThecreationofpublickeycryptographybyDi?eandHellmanin1976andthe subsequent invention of the RSA public key cryptosystem by Rivest, Shamir, and Adleman in 1978 are watershed events in the long history of secret c- munications. It is hard to overestimate the importance of public key cr- tosystems and their associated digital signature schemes in the modern world of computers and the Internet. This book provides an introduction to the theory of public key cryptography and to the mathematical ideas underlying that theory. Public key cryptography draws on many areas of mathematics, including number theory, abstract algebra, probability, and information theory. Each of these topics is introduced and developed in su?cient detail so that this book provides a self-contained course for the beginning student. The only prerequisite is a ?rst course in linear algebra. On the other hand, students with stronger mathematical backgrounds can move directly to cryptographic applications and still have time for advanced topics such as elliptic curve pairings and lattice-reduction algorithms. Amongthemanyfacetsofmoderncryptography,thisbookchoosestoc- centrate primarily on public key cryptosystems and digital signature schemes. This allows for an in-depth development of the necessary mathematics - quired for both the construction of these schemes and an analysis of their security. The reader who masters the material in this book will not only be well prepared for further study in cryptography, but will have acquired a real understanding of the underlying mathematical principles on which modern cryptography is based.
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Discrete Logarithms and DiffieHellman
Integer Factorization and RSA
Combinatorics Probability and Information Theory
Elliptic Curves and Cryptography
Lattices and Cryptography
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Alice and Bob Alice’s approximately B-smooth Babai’s bits Bob’s Chinese remainder theorem chooses ciphertext ciphertext letter coefficients compute congruence cryptography decryption Definition described discrete logarithm problem E(Fp ECDLP elements ElGamal elliptic curve encryption ephemeral key equal equation Euclidean algorithm example Exercise exponent extended Euclidean algorithm factor Fermat’s little theorem finite field formula gcd(a inverse keyword KPub large number lattice linear LLL algorithm mathematical matrix method Miller–Rabin mod q modulo multiplication nonzero NTRU pairing plaintext letter Pollard’s polynomial Pr(E Pr(F prime number primitive root private key probability proof Proposition Prove public key cryptosystem quadratic random variable randomly Remark ring root modulo Samantha satisfying secret Section sends sieve solution solve the discrete square modulo square root steps subset-sum substitution cipher Suppose Table theory vector verify Vigen`ere Weil pairing