An Introduction to Mathematical Cryptography (2nd Edition) - download pdf or read online

By Joseph H. Silverman, Jeffrey Hoffstein, Jill Pipher

ISBN-10: 1493917110

ISBN-13: 9781493917112

This self-contained creation to trendy cryptography emphasizes the math in the back of the speculation of public key cryptosystems and electronic signature schemes. The ebook makes a speciality of those key issues whereas constructing the mathematical instruments wanted for the development and safeguard research of various cryptosystems. in basic terms uncomplicated linear algebra is needed of the reader; strategies from algebra, quantity idea, and chance are brought and built as required. this article presents a fantastic advent for arithmetic and computing device technology scholars to the mathematical foundations of recent cryptography. The publication comprises an in depth bibliography and index; supplementary fabrics can be found online.

The publication covers quite a few subject matters which are thought of relevant to mathematical cryptography. Key subject matters include:

* classical cryptographic buildings, reminiscent of Diffie–Hellmann key alternate, discrete logarithm-based cryptosystems, the RSA cryptosystem, and electronic signatures;

* basic mathematical instruments for cryptography, together with primality checking out, factorization algorithms, likelihood concept, details conception, and collision algorithms;

* an in-depth therapy of vital cryptographic strategies, equivalent to elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem.

The moment variation of An advent to Mathematical Cryptography encompasses a major revision of the fabric on electronic signatures, together with an past creation to RSA, Elgamal, and DSA signatures, and new fabric on lattice-based signatures and rejection sampling. Many sections were rewritten or improved for readability, specifically within the chapters on details concept, elliptic curves, and lattices, and the bankruptcy of extra themes has been increased to incorporate sections on electronic money and homomorphic encryption. a variety of new workouts were integrated.

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

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6 modulo 7. Hence a6 ≡ 1 0 (mod 7) (mod 7) if 7 a, if 7 | a. Further experiments with other primes suggest that this example reflects a general fact. 24 (Fermat’s Little Theorem). Let p be a prime number and let a be any integer. Then ap−1 ≡ 1 (mod p) 0 (mod p) if p a, if p | a. Proof. There are many proofs of Fermat’s little theorem. If you have studied group theory, the quickest proof is to observe that the nonzero elements in Fp form a group F∗p of order p − 1, so by Lagrange’s theorem, every element of F∗p has order dividing p − 1.

See [66, page 441] for a brief discussion and further references. 26 1. An Introduction to Cryptography Step 3. Compute g A (mod N ) using the formula g A = g A0 +A1 ·2+A2 ·2 2 +A3 ·23 +···+Ar ·2r 2 3 r = g A0 · (g 2 )A1 · (g 2 )A2 · (g 2 )A3 · · · (g 2 )Ar A1 A2 A3 Ar 0 ≡ aA 0 · a1 · a2 · a3 · · · ar (mod N ). 4) Note that the quantities a0 , a1 , . . , ar were computed in Step 2. 4) can be computed by looking up the values of the ai ’s whose exponent Ai is 1 and then multiplying them together.

We denote this value of b by a−1 mod p, or if p has already been specified, then simply by a−1 . 5. Powers and Primitive Roots in Finite Fields 29 Proof. 13(b) using the prime modulus p, since if a ∈ Z/pZ is not zero, then gcd(a, p) = 1. 22. 11) gives us an efficient computational method for computing a−1 mod p. We simply solve the equation au + pv = 1 in integers u and v, and then u = a−1 mod p. 26. 21 can be restated by saying that if p is prime, then (Z/pZ)∗ = {1, 2, 3, 4, . . , p − 1}. In other words, when the 0 element is removed from Z/pZ, the remaining elements are units and closed under multiplication.

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

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