An Introduction to Mathematical Cryptography by Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman

By Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman

This self-contained advent to fashionable cryptography emphasizes the math in the back of the speculation of public key cryptosystems and electronic signature schemes. The e-book specializes in those key subject matters whereas constructing the mathematical instruments wanted for the development and safety research of numerous cryptosystems. merely uncomplicated linear algebra is needed of the reader; concepts from algebra, quantity concept, and chance are brought and constructed as required. this article offers a fantastic advent for arithmetic and machine 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 a number of themes which are thought of significant to mathematical cryptography. Key issues include:

  • classical cryptographic buildings, equivalent to DiffieHellmann key trade, discrete logarithm-based cryptosystems, the RSA cryptosystem, and electronic signatures;
  • fundamental mathematical instruments for cryptography, together with primality checking out, factorization algorithms, likelihood concept, details conception, and collision algorithms;
  • an in-depth remedy of vital cryptographic concepts, equivalent to elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem.

The moment variation of An advent to Mathematical Cryptography encompasses a major revision of the fabric 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 accelerated for readability, specifically within the chapters on details idea, elliptic curves, and lattices, and the bankruptcy of extra themes has been elevated to incorporate sections on electronic funds and homomorphic encryption. various new routines were included.

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More generally, any equation Au + Bv = gcd(A, B) can be reduced to the case of relatively prime numbers by dividing both sides by gcd(A, B). Thus B A u+ v = 1, gcd(A, B) gcd(A, B) where a = A/ gcd(A, B) and b = B/ gcd(A, B) are relatively prime and satisfy au+bv = 1. For example, we found earlier that 2024 and 748 have greatest common divisor 44 and satisfy −7 · 2024 + 19 · 748 = 44. Dividing both sides by 44, we obtain −7 · 46 + 19 · 17 = 1. Thus 2024/44 = 46 and 748/44 = 17 are relatively prime, and u = −7 and v = 19 are the coefficients of a linear combination of 46 and 17 that equals 1.

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. 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.

13(b) tells us that a has an inverse modulo m if and only if gcd(a, m) = 1. Numbers that have inverses are called units. We denote the set of all units by (Z/mZ)∗ = {a ∈ Z/mZ : gcd(a, m) = 1} = {a ∈ Z/mZ : a has an inverse modulo m}. The set (Z/mZ)∗ is called the group of units modulo m. Notice that if a1 and a2 are units modulo m, then so is a1 a2 . ) So when we multiply two units, we always get a unit. On the other hand, if we add two units, we often do not get a unit. 17. The group of units modulo 24 is (Z/24Z)∗ = {1, 5, 7, 11, 13, 17, 19, 23}.

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