An Introduction to Mathematical Cryptography (2nd Edition) by Joseph H. Silverman, Jeffrey Hoffstein, Jill Pipher

By Joseph H. Silverman, Jeffrey Hoffstein, Jill Pipher

This self-contained advent to fashionable cryptography emphasizes the maths in the back of the speculation of public key cryptosystems and electronic signature schemes. The ebook specializes in those key issues whereas constructing the mathematical instruments wanted for the development and safety research of numerous cryptosystems. in simple terms uncomplicated linear algebra is needed of the reader; concepts from algebra, quantity idea, and likelihood are brought and built as required. this article offers an awesome advent for arithmetic and desktop technological know-how scholars to the mathematical foundations of contemporary cryptography. The publication contains an in depth bibliography and index; supplementary fabrics can be found online.

The booklet covers a number of subject matters which are thought of relevant to mathematical cryptography. Key subject matters include:

* classical cryptographic buildings, equivalent to Diffie–Hellmann key trade, discrete logarithm-based cryptosystems, the RSA cryptosystem, and electronic signatures;

* primary mathematical instruments for cryptography, together with primality trying out, factorization algorithms, likelihood idea, details idea, and collision algorithms;

* an in-depth therapy of significant cryptographic suggestions, reminiscent of elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem.

The moment version of An advent to Mathematical Cryptography features a major revision of the fabric on electronic signatures, together with an prior advent to RSA, Elgamal, and DSA signatures, and new fabric on lattice-based signatures and rejection sampling. Many sections were rewritten or increased for readability, in particular within the chapters on info concept, elliptic curves, and lattices, and the bankruptcy of extra issues has been extended to incorporate sections on electronic money and homomorphic encryption. a variety of new routines were incorporated.

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Extra resources for An Introduction to Mathematical Cryptography (2nd Edition) (Undergraduate Texts in Mathematics)

Example text

Suppose now that we want to find the greatest common divisor of a and b. We first divide a by b to get a=b·q+r with 0 ≤ r < b. 1) If d is any common divisor of a and b, then it is clear from Eq. 1) that d is also a divisor of r. 1) shows that e is a divisor of a. In other words, the common divisors of a and b are the same as the common divisors of b and r; hence gcd(a, b) = gcd(b, r). We repeat the process, dividing b by r to get another quotient and remainder, say with 0 ≤ r′ < r. b = r · q ′ + r′ Then the same reasoning shows that gcd(b, r) = gcd(r, r′ ).

Then a · b ≡ 1 (mod m) for some integer b if and only if gcd(a, m) = 1. Further, if a · b1 ≡ a · b2 ≡ 1 (mod m), then b1 ≡ b2 (mod m). We call b the (multiplicative) inverse of a modulo m. Proof. 15. (b) Suppose first that gcd(a, m) = 1. 11 tells us that we can find integers u and v satisfying au + mv = 1. This means that au − 1 = −mv is divisible by m, so by definition, au ≡ 1 (mod m). In other words, we can take b = u. For the other direction, suppose that a has an inverse modulo m, say a · b ≡ 1 (mod m).

4 illustrates the ring Z/5Z by giving complete addition and multiplication tables modulo 5. 22 1. 16. If you have studied ring theory, you will recognize that Z/mZ is the quotient ring of Z by the principal ideal mZ, and that the numbers 0, 1, . . , m − 1 are actually coset representatives for the congruence classes that comprise the elements of Z/mZ. For a discussion of congruence classes and general quotient rings, see Sect. 2. Definition. 13(b) tells us that a has an inverse modulo m if and only if gcd(a, m) = 1.

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