Read e-book online 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 advent to trendy cryptography emphasizes the math in the back of the idea of public key cryptosystems and electronic signature schemes. The e-book makes a speciality of those key issues whereas constructing the mathematical instruments wanted for the development and safety research of various cryptosystems. in simple terms uncomplicated linear algebra is needed of the reader; ideas from algebra, quantity thought, and likelihood are brought and constructed as required. this article presents a fantastic creation for arithmetic and machine technological know-how scholars to the mathematical foundations of recent cryptography. The booklet contains an intensive bibliography and index; supplementary fabrics can be found online.

The publication covers a number of themes which are thought of imperative to mathematical cryptography. Key subject matters include:

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

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

* an in-depth remedy of significant cryptographic thoughts, comparable to elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem.

The moment version of An creation to Mathematical Cryptography features 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 elevated for readability, specially within the chapters on info concept, elliptic curves, and lattices, and the bankruptcy of extra subject matters has been increased to incorporate sections on electronic money and homomorphic encryption. various new workouts were integrated.

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

Sample text

Using that secret key, they can both encrypt and decrypt messages, so Bob and Alice have equal (or symmetric) knowledge and abilities. For this reason, ciphers of this sort are known as symmetric ciphers. , a set) of possible keys K to encrypt a plaintext message m chosen from a space of possible messages M, and the result of the encryption process is a ciphertext c belonging to a space of possible ciphertexts C. Thus encryption may be viewed as a function e:K×M→C whose domain K×M is the set of pairs (k, m) consisting of a key k and a plaintext m and whose range is the space of ciphertexts C.

Here m0 , m1 , . . , mB−1 are each 0 or 1. Similarly, we identify the key space K and the ciphertext space C with sets of integers corresponding to bit strings of a certain blocksize. For notational convenience, we denote the blocksizes for keys, plaintexts, and ciphertexts by Bk , Bm , and Bc . They need not be the same. Thus we have identified K, M, and C with sets of positive integers K = {k ∈ Z : 0 ≤ k < 2Bk }, M = {m ∈ Z : 0 ≤ m < 2Bm }, C = {c ∈ Z : 0 ≤ c < 2Bc }. An important question immediately arises: how large should Alice and Bob make the set K, or equivalently, how large should they choose the key blocksize Bk ?

We have now proven our claim that ri+2 < 12 ri for all i. Using this inequality repeatedly, we find that r2k+1 < 1 1 1 1 1 1 r2k−1 < r2k−3 < r2k−5 < r2k−7 < · · · < k r1 = k b. 2 4 8 16 2 2 Hence if 2k ≥ b, then r2k+1 < 1, which forces r2k+1 to equal 0 and the algorithm to terminate. In terms of Fig. 2. Divisibility and Greatest Common Divisors 15 t + 1 ≤ 2k + 1, and thus t ≤ 2k. Further, there are exactly t divisions performed in Fig. 2, so the Euclidean algorithm terminates in at most 2k iterations.

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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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