By Joseph H. Silverman, Jeffrey Hoffstein, Jill Pipher
This self-contained creation to fashionable cryptography emphasizes the maths at the back of the idea of public key cryptosystems and electronic signature schemes. The booklet makes a speciality of those key issues whereas constructing the mathematical instruments wanted for the development and safety research of numerous cryptosystems. simply simple linear algebra is needed of the reader; options from algebra, quantity thought, and chance are brought and built as required. this article presents an excellent creation for arithmetic and laptop technological know-how scholars to the mathematical foundations of recent cryptography. The ebook comprises an intensive bibliography and index; supplementary fabrics can be found online.
The booklet covers numerous issues which are thought of critical to mathematical cryptography. Key themes include:
* classical cryptographic buildings, resembling Diffie–Hellmann key alternate, discrete logarithm-based cryptosystems, the RSA cryptosystem, and electronic signatures;
* primary mathematical instruments for cryptography, together with primality trying out, factorization algorithms, chance idea, info concept, and collision algorithms;
* an in-depth remedy of vital cryptographic suggestions, reminiscent of 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 fabric on electronic signatures, together with an prior 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, in particular within the chapters on details concept, elliptic curves, and lattices, and the bankruptcy of extra subject matters has been increased to incorporate sections on electronic funds and homomorphic encryption. a number of new routines were integrated.
Read or Download An Introduction to Mathematical Cryptography (2nd Edition) (Undergraduate Texts in Mathematics) PDF
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Additional resources for An Introduction to Mathematical Cryptography (2nd Edition) (Undergraduate Texts in Mathematics)
28. 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.
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.
24) tells us that 215485862 ≡ 1 (mod 15485863). Thus without doing any computing, we know that the number 215485862 − 1, a number having more than two million digits, is a multiple of 15485863. 26. 24) and the fast powering algorithm (Sect. 2) provide us with a reasonably eﬃcient method of computing inverses modulo p, namely a−1 ≡ ap−2 (mod p). This congruence is true because if we multiply ap−2 by a, then Fermat’s theorem tells us that the product is equal to 1 modulo p. 22. 3, Theorem E]. 27.
An Introduction to Mathematical Cryptography (2nd Edition) (Undergraduate Texts in Mathematics) by Joseph H. Silverman, Jeffrey Hoffstein, Jill Pipher