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

Posted by

By Joseph H. Silverman, Jeffrey Hoffstein, Jill Pipher

This self-contained advent to fashionable cryptography emphasizes the math in the back of the idea of public key cryptosystems and electronic signature schemes. The ebook specializes in those key themes whereas constructing the mathematical instruments wanted for the development and defense research of numerous cryptosystems. in basic terms simple linear algebra is needed of the reader; ideas from algebra, quantity conception, and likelihood are brought and constructed as required. this article presents a terrific creation for arithmetic and computing device technological know-how scholars to the mathematical foundations of recent cryptography. The booklet comprises an intensive bibliography and index; supplementary fabrics can be found online.

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

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

* primary mathematical instruments for cryptography, together with primality trying out, factorization algorithms, chance concept, details idea, and collision algorithms;

* an in-depth therapy of vital cryptographic thoughts, similar 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 encompasses a major revision of the cloth on electronic signatures, together with an past creation to RSA, Elgamal, and DSA signatures, and new fabric on lattice-based signatures and rejection sampling. Many sections were rewritten or increased for readability, specifically within the chapters on details concept, elliptic curves, and lattices, and the bankruptcy of extra issues has been multiplied to incorporate sections on electronic money and homomorphic encryption. quite a few new workouts were incorporated.

Show description

Read Online or Download An Introduction to Mathematical Cryptography (2nd Edition) (Undergraduate Texts in Mathematics) PDF

Best cryptography books

Disappearing Cryptography: Information Hiding: Steganography & Watermarking (3rd Edition) (The Morgan Kaufmann Series in Software Engineering and Programming)

Cryptology is the perform of hiding electronic details by way of a number of obfuscatory and steganographic suggestions. the applying of acknowledged recommendations enables message confidentiality and sender/receiver id authentication, and is helping to make sure the integrity and safety of laptop passwords, ATM card info, electronic signatures, DVD and HDDVD content material, and digital trade.

Hieroglyphs: A Very Short Introduction (Very Short Introductions)

Hieroglyphs have been way over a language. They have been an omnipresent and omnipotent strength in speaking the messages of historic Egyptian tradition for over 3 thousand years. This historic type of expression was once used as artwork, as a way of selecting Egyptian-ness, even for conversation with the gods.

SSCP Systems Security Certified Practitioner All-in-One Exam Guide, Second Edition

This fully-updated, built-in self-study method bargains whole assurance of the revised 2015 structures safety qualified Practitioner (SSCP) examination domain names completely revised for the April 2015 examination replace, SSCP platforms safety qualified Practitioner All-in-One examination advisor, moment version permits you to take the examination with entire self belief.

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

Sample text

Elements with this property are called primitive roots of Fp or generators of F∗p . They are the elements of F∗p having order p − 1. Proof. See [137, Chapter 20] or one of the texts [35, 52, 59, 100, 111]. 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.

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.

The inverse of 2 modulo 5 is 3, since 2·3 ≡ 1 (mod 5), so 2−1 ≡ 3 (mod 5). Similarly gcd(4, 15) = 1 so 4−1 exists modulo 15. In fact 4 · 4 ≡ 1 (mod 15) so 4 is its own inverse modulo 15. We can even work with fractions a/d modulo m as long as the denominator is relatively prime to m. For example, we can compute 5/7 modulo 11 by first observing that 7 · 8 ≡ 1 (mod 11), so 7−1 ≡ 8 (mod 11). Then 5 = 5 · 7−1 ≡ 5 · 8 ≡ 40 ≡ 7 (mod 11). 15. In the preceding examples it was easy to find inverses modulo m by trial and error.

Download PDF sample

Rated 4.80 of 5 – based on 35 votes