By Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman
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 e-book makes a speciality of those key themes whereas constructing the mathematical instruments wanted for the development and safeguard research of various cryptosystems. merely simple linear algebra is needed of the reader; thoughts from algebra, quantity thought, and chance are brought and constructed as required. this article offers a fantastic creation for arithmetic and computing device technological know-how scholars to the mathematical foundations of contemporary cryptography. The booklet comprises an intensive bibliography and index; supplementary fabrics can be found online.
The e-book covers quite a few themes which are thought of valuable to mathematical cryptography. Key subject matters include:
- classical cryptographic buildings, resembling Diffie–Hellmann key trade, discrete logarithm-based cryptosystems, the RSA cryptosystem, and electronic signatures;
- fundamental mathematical instruments for cryptography, together with primality trying out, factorization algorithms, chance idea, info thought, and collision algorithms;
- an in-depth remedy of vital cryptographic concepts, comparable 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 contains a major revision of the cloth on electronic signatures, together with an prior advent to RSA, Elgamal, and DSA signatures, and new fabric on lattice-based signatures and rejection sampling. Many sections were rewritten or improved for readability, specifically within the chapters on details conception, elliptic curves, and lattices, and the bankruptcy of extra themes has been elevated to incorporate sections on electronic funds and homomorphic encryption. quite a few new routines were included.
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Additional resources for An Introduction to Mathematical Cryptography
This means that au − 1 = −mv is divisible by m, so by deﬁnition, au ≡ 1 (mod m). In other words, we can take b = u. For the other direction, suppose that a has an inverse modulo m, say a · b ≡ 1 (mod m). This means that ab − 1 = cm for some integer c. It follows that gcd(a, m) divides ab − cm = 1, so gcd(a, m) = 1. This completes the proof that a has an inverse modulo m if and only if gcd(a, m) = 1. It remains to show that the inverse is unique modulo m. So suppose that a · b1 ≡ a · b2 ≡ 1 (mod m).
Similarly, decryption is a function d : K × C → M. Of course, we want the decryption function to “undo” the results of the encryption function. Mathematically, this is expressed by the formula d k, e(k, m) = m for all k ∈ K and all m ∈ M. It is sometimes convenient to write the dependence on k as a subscript. Then for each key k, we get a pair of functions ek : M −→ C and dk : C −→ M satisfying the decryption property dk ek (m) = m for all m ∈ M. 38 1. An Introduction to Cryptography In other words, for every key k, the function dk is the inverse function of the function ek .
Then a · b ≡ 1 (mod m) for some integer b if and only if gcd(a, m) = 1. Further, if a · b1 ≡ a · b2 ≡ 1 (mod m), then b1 ≡ b2 (mod m). We call b the (multiplicative) inverse of a modulo m. Proof. 15. (b) Suppose ﬁrst that gcd(a, m) = 1. 11 tells us that we can ﬁnd integers u and v satisfying au + mv = 1. This means that au − 1 = −mv is divisible by m, so by deﬁnition, au ≡ 1 (mod m). In other words, we can take b = u. For the other direction, suppose that a has an inverse modulo m, say a · b ≡ 1 (mod m).
An Introduction to Mathematical Cryptography by Jeffrey Hoffstein, Jill Pipher, Joseph H. Silverman