Mathematical Foundations of Public Key Cryptography by Xiaoyun Wang, Guangwu Xu, Mingqiang Wang, Xianmeng Meng

By Xiaoyun Wang, Guangwu Xu, Mingqiang Wang, Xianmeng Meng

In Mathematical Foundations of Public Key Cryptography, the authors combine the result of greater than twenty years of analysis and educating event to assist scholars bridge the distance among math conception and crypto perform. The publication presents a theoretical constitution of basic quantity thought and algebra wisdom helping public-key cryptography.

Rather than just combining quantity thought and sleek algebra, this textbook gains the interdisciplinary features of cryptography―revealing the integrations of mathematical theories and public-key cryptographic functions. Incorporating the complexity idea of algorithms all through, it introduces the elemental quantity theoretic and algebraic algorithms and their complexities to supply a initial knowing of the purposes of mathematical theories in cryptographic algorithms.

Supplying a unbroken integration of cryptography and arithmetic, the e-book comprises insurance of basic quantity idea; algebraic constitution and attributes of crew, ring, and box; cryptography-related computing complexity and simple algorithms, in addition to lattice and basic equipment of lattice cryptanalysis.

The textual content involves eleven chapters. uncomplicated thought and instruments of user-friendly quantity thought, equivalent to congruences, primitive roots, residue sessions, and persevered fractions, are lined in Chapters 1-6. the fundamental options of summary algebra are brought in Chapters 7-9, the place 3 uncomplicated algebraic buildings of teams, earrings, and fields and their homes are defined.

Chapter 10 is ready computational complexities of a number of similar mathematical algorithms, and difficult difficulties reminiscent of integer factorization and discrete logarithm. bankruptcy eleven offers the fundamentals of lattice thought and the lattice foundation aid algorithm―the LLL set of rules and its program within the cryptanalysis of the RSA algorithm.

Containing a few workouts on key algorithms, the ebook is acceptable to be used as a textbook for undergraduate scholars and first-year graduate scholars in details defense courses. it's also a fantastic reference booklet for cryptography pros seeking to grasp public-key cryptography.

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The result is proved. 43 Let a be a positive integer and τ(a) denote the number of positive factors of a (usually called the divisor function). 5), then τ(a) = (α1 + 1) · · · (αs + 1) = τ(pα11 ) · · · τ(pαss ). 44 Let integer a ≥ 2. 1 1. If a is a composite, then there is a prime p|a and p ≤ a 2 . 2. 4, then there is a prime p|a and 1 p ≤ as . 44 gives a method for finding prime—the Sieve of Eratosthenes. The Sieve of Eratosthenes is used to list all primes that are less than or equal to a positive integer n.

If a = 2, the result is true as 2 is a prime. Assume that the result is true for 2 ≤ a < n. Consider the case that a = n. The result is obviously true if n is prime. If n is composite, then n = n1 n2 for 2 ≤ n1 , n2 < n. By our induction hypothesis, n1 , n2 are both products of prime numbers: n1 = p11 · · · p1s , n2 = p21 · · · p2r . Therefore, a is a product of primes a = n = n1 n2 = p11 · · · p1s p21 · · · p2r . That proves the existence of prime factorization of integers. Uniqueness: If there are two forms of prime factorization a = p1 p2 · · · ps , p1 ≤ p2 ≤ · · · ≤ ps , a = q1 q2 · · · qr , q1 ≤ q2 ≤ · · · ≤ qr , where pi (1 ≤ i ≤ s), qi (1 ≤ i ≤ r) are primes.

In general, if m = m1 · · · mk , x = x(1) + m1 x(2) + · · · + m1 m2 · · · mk−1 x(k) , then if x(1) , · · ·, x(k) run over complete systems of residues modulo m1 , · · ·, mk , respectively, x runs over a complete system of residues modulo m. Proof We first show that the result is true for k = 2. In this case, there are m = m1 m2 numbers of xij , and we only need to prove that they are mutually incongruent. If (1) (2) (1) (2) (mod m1 m2 ), (1) (1) xi1 + m1 xj1 = xi1 j1 ≡ xi2 j2 = xi2 + m1 xj2 (1) (1) then xi1 = xi2 (mod m1 ).

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