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* offers an creation to public key cryptography and underlying arithmetic that's required for the topic. all the 8 chapters expands on a particular quarter of mathematical cryptography and gives an intensive record of exercises.

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

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**Sample text**

5. In many of the cryptosystems that we will study, it is important to know how many elements are in the unit group modulo m. This quantity is suﬃciently ubiquitous that we give it a name. Deﬁnition. Euler’s phi function (also sometimes known as Euler’s totient function) is the function φ(m) deﬁned by the rule ∗ φ(m) = # (Z/mZ) = #{0 ≤ a < m : gcd(a, m) = 1}. 18 that φ(24) = 8 and φ(7) = 6. 3. 1 works by shifting each letter in the alphabet a ﬁxed number of letters. 7. Then a shift cipher with shift k takes a plaintext letter corresponding to the number p and assigns it to the ciphertext letter corresponding to the number p + k mod 26.

Rounding this oﬀ gives r = 15909. After performing the Euclidean algorithm on two numbers, we can work our way back up the process to obtain an extremely interesting formula. Before giving the general result, we illustrate with an example. 10. 6 we used the Euclidean algorithm to compute gcd(2024, 748) as follows: 2024 = 748 · 2 + 528 748 = 528 · 1 + 220 528 = 220 · 2 + 88 220 = 88 · 2 + 44 88 = 44 · 2 + 0 ← gcd = 44 16 1. An Introduction to Cryptography We let a = 2024 and b = 748, so the ﬁrst line says that 528 = a − 2b.

The ﬁrst step is to divide a by b on your calculator, which will give a real number. Throw away the part after the decimal point to get the quotient q. Then the remainder r can be computed as r = a − b · q. For example, let a = 2387187 and b = 27573. 57697748, so q = 86 and r = a − b · q = 2387187 − 27573 · 86 = 15909. If you need just the remainder, you can instead take the decimal part (also sometimes called the fractional part) of a/b and multiply it by b. 00005604. Rounding this oﬀ gives r = 15909.

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

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