Get An Introduction to Mathematical Cryptography PDF

Get An Introduction to Mathematical Cryptography PDF

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 of the 8 chapters expands on a selected zone of mathematical cryptography and gives an in depth 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 booklet can be used as a self-study. This publication additionally presents a self-contained therapy of mathematical cryptography for the reader with restricted mathematical background.

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Extra resources for An Introduction to Mathematical Cryptography

Example text

18. The group of units modulo 7 is (Z/7Z)∗ = {1, 2, 3, 4, 5, 6}, since every number between 1 and 6 is relatively prime to 7. 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. 3. Modular arithmetic 19 In general, if a and b are relatively prime and if q1 , q2 , . . 2 on page 13, then the box has the form 0 1 q1 1 P1 0 Q1 q2 P2 Q2 ... ... qt−1 Pt−1 Qt−1 qt a b The entries in the box are calculated using the initial values P 1 = q1 , P2 = q2 · P1 + 1, Q1 = 1, Q2 = q2 · Q1 , and then, for i ≥ 3, using the formulas Pi = qi · Pi−1 + Pi−2 and Qi = qi · Qi−1 + Qi−2 . The final four entries in the box satisfy a · Qt−1 − b · Pt−1 = (−1)t . Multiplying both sides by (−1)t gives the solution u = (−1)t Qt−1 and v = (−1)t+1 Pt−1 to the equation au + bv = 1.

In particular, the order of a divides p − 1. prime factorization of m is m = 15485207 = 3853 · 4019. earlier defined the order of p in a to be the exponent of p when a is factored into primes. Thus unfortunately, the word “order” has two different meanings. You will need to judge which one is meant from the context. 5. Powers and primitive roots in finite fields 33 Proof. Let k be the order of a modulo p, so by definition ak ≡ 1 (mod p), and k is the smallest positive exponent with this property. We are given that an ≡ 1 (mod p).

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

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