Download e-book for kindle: An Introduction to Mathematical Cryptography by Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman

By Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman

ISBN-10: 0387779930

ISBN-13: 9780387779935

ISBN-10: 0387779949

ISBN-13: 9780387779942

An creation to Mathematical Cryptography presents an creation to public key cryptography and underlying arithmetic that's required for the topic. all the 8 chapters expands on a selected quarter of mathematical cryptography and offers an in depth checklist of exercises.

It is an acceptable textual content for complex scholars in natural and utilized arithmetic and machine technological know-how, or the e-book can be utilized as a self-study. This e-book additionally presents a self-contained therapy of mathematical cryptography for the reader with restricted mathematical background.

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Additional resources for An Introduction to Mathematical Cryptography

Example text

Of course, there is also the question of identifying a number as prime or composite. There are efficient tests that do this, even for very large numbers. 4. 42 1. An Introduction to Cryptography It is clear that Eve has a hard time guessing k, since there are approximately 2160 possibilities from which to choose. Is it also difficult for Eve to recover k if she knows the ciphertext c? The answer is yes, it is still difficult. Notice that the encryption function ek : M −→ C is surjective (onto) for any choice of key k.

Definition. 21) says that in the factorization of a positive integer a into primes, each prime p appears to a particular power. We denote this power by ordp (a) and call it the order (or exponent) of p in a. ) For example, the factorization of 1728 is 1728 = 26 · 33 , so ord2 (1728) = 6, ord3 (1728) = 3, and ordp (1728) = 0 for all primes p ≥ 5. Using the ordp notation, the factorization of a can be succinctly written as pordp (a) . a= primes p Note that this product makes sense, since ordp (a) is zero for all but finitely many primes.

Given one or more ciphertexts c1 , c2 , . . , cn ∈ C encrypted using the key k ∈ K, it must be very difficult to compute any of the corresponding plaintexts dk (c1 ), . . , dk (cn ) without knowledge of k. There is a fourth property that is desirable, although it is more difficult to achieve. 4. Given one or more pairs of plaintexts and their corresponding ciphertexts, (m1 , c1 ), (m2 , c2 ), . . , (mn , cn ), it must be difficult to decrypt any ciphertext c that is not in the given list without knowing k.

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An Introduction to Mathematical Cryptography by Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman

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