By Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman
This self-contained creation to fashionable cryptography emphasizes the math in the back of the idea of public key cryptosystems and electronic signature schemes. The e-book makes a speciality of those key subject matters whereas constructing the mathematical instruments wanted for the development and safeguard research of various cryptosystems. basically uncomplicated linear algebra is needed of the reader; options from algebra, quantity concept, and likelihood are brought and constructed as required. this article presents an awesome creation for arithmetic and laptop technological know-how scholars to the mathematical foundations of contemporary cryptography. The ebook comprises an intensive bibliography and index; supplementary fabrics can be found online.
The booklet covers quite a few subject matters which are thought of significant to mathematical cryptography. Key themes include:
- classical cryptographic buildings, equivalent to 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 idea, details idea, and collision algorithms;
- an in-depth remedy of significant cryptographic techniques, akin 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 features a major revision of the cloth on electronic signatures, together with an previous advent 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 info concept, elliptic curves, and lattices, and the bankruptcy of extra subject matters has been improved to incorporate sections on electronic money and homomorphic encryption. quite a few new workouts were included.
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Extra resources for An Introduction to Mathematical Cryptography
For example, if Alice and Bob use a simple substitution cipher, they should assume that Eve is aware of this fact. This illustrates a basic premise of modern cryptography called Kerckhoﬀ ’s principle, which says that the security of a cryptosystem should depend only on the secrecy of the key, and not on the secrecy of the encryption algorithm itself. If (K, M, C, e, d) is to be a successful cipher, it must have the following properties: 1. For any key k ∈ K and plaintext m ∈ M, it must be easy to compute the ciphertext ek (m).
Powers and Primitive Roots in Finite Fields 31 If p | a, then it is clear that every power of a is divisible by p. So we only need to consider the case that p a. , (p − 1)a reduced modulo p. 8) There are p − 1 numbers in this list, and we claim that they are all diﬀerent. To see why, take any two of them, say ja mod p and ka mod p, and suppose that they are the same. This means that ja ≡ ka (mod p), (j − k)a ≡ 0 and hence that (mod p). Thus p divides the product (j − k)a. 19 tells us that either p divides j − k or p divides a.
3. 1 Modular Arithmetic and Shift Ciphers Recall that the Caesar (or shift) cipher studied in Sect. 1 works by shifting each letter in the alphabet a ﬁxed number of letters. 7. 7: Assigning numbers to letters Then a shift cipher with shift k takes a plaintext letter corresponding to the number p and assigns it to the ciphertext letter corresponding to the number p + k mod 26. Notice how the use of modular arithmetic, in this case modulo 26, simpliﬁes the description of the shift cipher. The shift amount serves as both the encryption key and the decryption key.
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