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

By Joseph H. Silverman, Jeffrey Hoffstein, Jill Pipher

ISBN-10: 1493917110

ISBN-13: 9781493917112

This self-contained creation to trendy cryptography emphasizes the maths in the back of the speculation of public key cryptosystems and electronic signature schemes. The e-book makes a speciality of those key themes whereas constructing the mathematical instruments wanted for the development and safety research of various cryptosystems. purely simple linear algebra is needed of the reader; strategies from algebra, quantity conception, and chance are brought and built as required. this article offers an incredible advent for arithmetic and desktop technology 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 quite a few themes which are thought of vital to mathematical cryptography. Key themes include:

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

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

* an in-depth therapy of significant cryptographic techniques, 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 contains a major revision of the fabric on electronic signatures, together with an previous advent to RSA, Elgamal, and DSA signatures, and new fabric on lattice-based signatures and rejection sampling. Many sections were rewritten or accelerated for readability, specifically within the chapters on details idea, elliptic curves, and lattices, and the bankruptcy of extra themes has been accelerated to incorporate sections on electronic funds 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

Strings of exactly B ones and zeros. We call B the blocksize of the cipher. A general plaintext message then consists of a list of message blocks chosen from M, and the encryption function transforms the message blocks into a list of ciphertext blocks in C, where each block is a sequence of B bits. If the plaintext ends with a block of fewer than B bits, we pad the end of the block with zeros. Keep in mind that this encoding process, which converts the original plaintext message into a sequence of blocks of bits in M, is public knowledge.

In mathematical terms, this means that Eve knows the functions e and d. What Eve does not know is the particular key k that Alice and Bob are using. For example, if Alice and Bob use a simple substitution cipher, they should assume that Eve is aware of this fact. This illustrates a basic premise of modern cryptography called Kerckhoff ’s principle, which says that the security of a cryptosystem should depend only on the secrecy of the key, and not on the secrecy of the encryption algorithm itself.

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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