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