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

By Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman

This self-contained advent to trendy cryptography emphasizes the math in the back of the speculation of public key cryptosystems and electronic signature schemes. The booklet specializes in those key issues whereas constructing the mathematical instruments wanted for the development and protection research of various cryptosystems. purely uncomplicated linear algebra is needed of the reader; concepts from algebra, quantity conception, and chance are brought and built as required. this article presents a terrific creation for arithmetic and laptop technology scholars to the mathematical foundations of recent cryptography. The booklet comprises an in depth bibliography and index; supplementary fabrics can be found online.

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

  • classical cryptographic structures, similar to DiffieHellmann key alternate, discrete logarithm-based cryptosystems, the RSA cryptosystem, and electronic signatures;
  • fundamental mathematical instruments for cryptography, together with primality trying out, factorization algorithms, chance conception, details idea, and collision algorithms;
  • an in-depth therapy of vital cryptographic suggestions, resembling elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem.

The moment variation of An creation to Mathematical Cryptography incorporates a major revision of the cloth on electronic signatures, together with an past 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, in particular within the chapters on details concept, elliptic curves, and lattices, and the bankruptcy of extra subject matters has been improved to incorporate sections on electronic funds and homomorphic encryption. various new routines were included.

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Additional resources for An Introduction to Mathematical Cryptography

Example text

Notice that the proposition is false for composite numbers. For example, 6 divides 3 · 10, but 6 divides neither 3 nor 10. 9 Note that log (A) means the usual logarithm to the base 2, not the so-called discrete 2 logarithm that will be discussed in Chap. 2. 4. 19. Let p be a prime number, and suppose that p divides the product ab of two integers a and b. Then p divides at least one of a and b. More generally, if p divides a product of integers, say p | a1 a2 · · · an , then p divides at least one of the individual ai .

Consider the number m = 15485207. Using the powering algorithm, it is not hard to compute (on a computer) 2m−1 = 215485206 ≡ 4136685 (mod 15485207). We did not get the value 1, so it seems that Fermat’s little theorem is not true for m. What does that tell us? If m were prime, then Fermat’s little theorem says that we would have obtained 1. Hence the fact that we did not get 1 proves that the number m = 15485207 is not prime. Think about this for a minute, because it’s actually a bit astonishing.

Jumping forward several centuries, we note that the machine ciphers that played a large role in World War II were, in essence, extremely complicated polyalphabetic ciphers. Ciphers and codes14 for both political and military purposes become increasingly widespread during the eighteenth, nineteenth, and early twentieth centuries, as did cryptanalytic methods, although the level of sophistication varied widely from generation to generation and from country to country. S. Army was using ciphers, inferior to those invented in Italy in the 1600s, that any trained cryptanalyst of the time would have been able to break in a few hours!

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