By Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman
This self-contained advent to trendy cryptography emphasizes the maths at the back of the speculation of public key cryptosystems and electronic signature schemes. The ebook specializes in those key subject matters whereas constructing the mathematical instruments wanted for the development and defense research of numerous cryptosystems. merely uncomplicated linear algebra is needed of the reader; thoughts from algebra, quantity concept, and likelihood are brought and built as required. this article presents a terrific advent for arithmetic and computing device technological know-how scholars to the mathematical foundations of recent cryptography. The e-book contains an intensive bibliography and index; supplementary fabrics can be found online.
The ebook covers numerous themes which are thought of relevant to mathematical cryptography. Key subject matters include:
- classical cryptographic buildings, reminiscent of Diffie–Hellmann key trade, discrete logarithm-based cryptosystems, the RSA cryptosystem, and electronic signatures;
- fundamental mathematical instruments for cryptography, together with primality checking out, factorization algorithms, chance conception, info concept, and collision algorithms;
- an in-depth remedy of significant cryptographic options, equivalent to 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 encompasses a major revision of the cloth 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 improved for readability, specially within the chapters on info concept, elliptic curves, and lattices, and the bankruptcy of extra issues has been elevated to incorporate sections on electronic funds and homomorphic encryption. a variety of new workouts were included.
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Extra info for An Introduction to Mathematical Cryptography
He uses a secret key k to scramble his plaintext message m and turn it into a ciphertext c. Alice, upon receiving c, uses the secret key k to unscramble c and reconstitute m. If this procedure is to work properly, then both Alice and Bob must possess copies of the secret key k, and if the system is to provide security, then their adversary Eve must not know k, must not be able to guess k, and must not be able to recover m from c without knowing k. In this section we formulate the notion of a cryptosystem in abstract mathematical terms.
2. Divisibility and Greatest Common Divisors 15 t + 1 ≤ 2k + 1, and thus t ≤ 2k. Further, there are exactly t divisions performed in Fig. 2, so the Euclidean algorithm terminates in at most 2k iterations. Choose the smallest such k, so 2k ≥ b > 2k−1 . 7. 8. We proved that the Euclidean algorithm applied to a and b with a ≥ b requires no more than 2 log2 (b) + 2 iterations to compute gcd(a, b). This estimate can be somewhat improved. 14; see . 9. One way to compute quotients and remainders is by long division, as we did on page 12.
P − 1 in some mixed up order. Now consider what happens when we multiply together all of the numbers a, 2a, 3a, . . 8) and reduce the product modulo p. This is the same as multiplying together all of the numbers 1, 2, 3, . . , p − 1 modulo p, so we get a congruence a · 2a · 3a · · · (p − 1)a ≡ 1 · 2 · 3 · · · (p − 1) (mod p). There are p − 1 copies of a appearing on the left-hand side. We factor these out and use factorial notation (p − 1)! = 1 · 2 · · · (p − 1) to obtain ap−1 · (p − 1)! ≡ (p − 1)!
An Introduction to Mathematical Cryptography by Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman