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

By Joseph H. Silverman, Jeffrey Hoffstein, Jill Pipher

This self-contained advent to trendy cryptography emphasizes the math at 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 basic terms easy linear algebra is needed of the reader; concepts from algebra, quantity idea, and likelihood are brought and constructed as required. this article presents an excellent advent for arithmetic and computing device technological know-how scholars to the mathematical foundations of contemporary cryptography. The booklet comprises an in depth bibliography and index; supplementary fabrics can be found online.

The ebook covers numerous subject matters which are thought of relevant to mathematical cryptography. Key themes include:

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

* basic mathematical instruments for cryptography, together with primality checking out, factorization algorithms, chance concept, details conception, and collision algorithms;

* an in-depth therapy of vital cryptographic strategies, corresponding to 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 creation 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 idea, elliptic curves, and lattices, and the bankruptcy of extra subject matters has been multiplied to incorporate sections on electronic funds and homomorphic encryption. quite a few new workouts were incorporated.

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

Example text

38 1. 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.

Multiplying both sides by (−1)t gives the solution u = (−1)t Qt−1 and v = (−1)t+1 Pt−1 to the equation au + bv = 1. 3 Modular Arithmetic You may have encountered “clock arithmetic” in grade school, where after you get to 12, the next number is 1. This leads to odd-looking equations such as 6+9=3 and 2 − 3 = 11. These look strange, but they are true using clock arithmetic, since for example 11 o’clock is 3 h before 2 o’clock. So what we are really doing is first computing 2 − 3 = −1 and then adding 12 to the answer.

31. The field F11 has 2 as a primitive root, since in F11 , 20 = 1 21 = 2 22 = 4 23 = 8 24 = 5 25 = 10 26 = 9 27 = 7 28 = 3 29 = 6. Thus all 10 nonzero elements of F11 have been generated as powers of 2. On the other hand, 2 is not a primitive root for F17 , since in F17 , 20 = 1 21 = 2 22 = 4 23 = 8 25 = 15 26 = 13 27 = 9 28 = 1, 24 = 16 so we get back to 1 before obtaining all 16 nonzero values modulo 17. However, it turns out that 3 is a primitive root for 17, since in F17 , 34 1. An Introduction to Cryptography 30 = 1 31 = 3 32 = 9 33 = 10 36 = 15 37 = 11 38 = 16 39 = 14 312 = 4 313 = 12 314 = 2 34 = 13 310 = 8 35 = 5 311 = 7 315 = 6.

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