Information Theory

An Introduction to Mathematical Cryptography - download pdf or read online

By Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman

ISBN-10: 1493917102

ISBN-13: 9781493917105

ISBN-10: 1493917110

ISBN-13: 9781493917112

This self-contained advent to fashionable cryptography emphasizes the maths at the back of the idea of public key cryptosystems and electronic signature schemes. The e-book specializes in those key subject matters whereas constructing the mathematical instruments wanted for the development and defense research of various cryptosystems. in basic terms uncomplicated linear algebra is needed of the reader; ideas from algebra, quantity conception, and likelihood are brought and built as required. this article offers a great creation for arithmetic and laptop technological know-how scholars to the mathematical foundations of contemporary cryptography. The e-book contains an intensive bibliography and index; supplementary fabrics can be found online.

The publication covers numerous issues which are thought of imperative to mathematical cryptography. Key subject matters include:

  • classical cryptographic structures, corresponding to DiffieHellmann key alternate, discrete logarithm-based cryptosystems, the RSA cryptosystem, and electronic signatures;
  • fundamental mathematical instruments for cryptography, together with primality checking out, factorization algorithms, likelihood conception, details idea, and collision algorithms;
  • an in-depth remedy of vital 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 contains a major revision of the cloth 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 extended for readability, specifically within the chapters on details thought, elliptic curves, and lattices, and the bankruptcy of extra issues has been improved to incorporate sections on electronic funds and homomorphic encryption. quite a few new workouts were included.

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Extra resources for An Introduction to Mathematical Cryptography

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An Introduction to Cryptography In other words, for every key k, the function dk is the inverse function of the function ek . In particular, this means that ek must be one-to-one, since if ek (m) = ek (m ), then m = dk ek (m) = dk ek (m ) = m . It is safest for Alice and Bob to assume that Eve knows the encryption method that is being employed. 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.

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 ? If Bk is too small, then Eve can check every number from 0 to 2Bk − 1 until she finds Alice and Bob’s key.

We denote this power by ordp (a) and call it the order (or exponent) of p in a. ) For example, the factorization of 1728 is 1728 = 26 · 33 , so ord2 (1728) = 6, ord3 (1728) = 3, and ordp (1728) = 0 for all primes p ≥ 5. Using the ordp notation, the factorization of a can be succinctly written as pordp (a) . a= primes p Note that this product makes sense, since ordp (a) is zero for all but finitely many primes. It is useful to view ordp as a function ordp : {1, 2, 3, . } −→ {0, 1, 2, 3, . }. 31. We now observe that if p is a prime, then every nonzero number modulo p has a multiplicative inverse modulo p.

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An Introduction to Mathematical Cryptography by Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman


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