Download e-book for kindle: Cryptography Made Simple by Nigel Smart

By Nigel Smart

ISBN-10: 3319219359

ISBN-13: 9783319219356

ISBN-10: 3319219367

ISBN-13: 9783319219363

In this introductory textbook the writer explains the major issues in cryptography. he's taking a contemporary process, the place defining what's intended by means of "secure" is as very important as growing anything that achieves that aim, and protection definitions are valuable to the dialogue throughout.

The chapters partially 1 provide a short creation to the mathematical foundations: modular mathematics, teams, finite fields, and likelihood; primality trying out and factoring; discrete logarithms; elliptic curves; and lattices. half 2 of the e-book indicates how ancient ciphers have been damaged, hence motivating the layout of contemporary cryptosystems because the Nineteen Sixties; this half additionally incorporates a bankruptcy on information-theoretic defense. half three covers the center facets of recent cryptography: the definition of safety; smooth flow ciphers; block ciphers and modes of operation; hash services, message authentication codes, and key derivation capabilities; the "naive" RSA set of rules; public key encryption and signature algorithms; cryptography in line with computational complexity; and certificate, key delivery and key contract. ultimately, half four addresses complicated prot ocols, the place the events could have various or perhaps conflicting safeguard pursuits: mystery sharing schemes; commitments and oblivious move; zero-knowledge proofs; and safe multi-party computation.

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MODULAR ARITHMETIC, GROUPS, FINITE FIELDS AND PROBABILITY Using this law with the following additional formulae gives rise to a recursive algorithm for the Legendre symbol: q p q·r p 2 p (2) (3) (4) = = q (mod p) p q r · , p p = (−1)(p 2 −1)/8 , . Assuming we can factor, we can now compute the Legendre symbol 15 17 = = = 3 5 · 17 17 17 17 · 3 5 2 2 · 3 5 by equation (3) by equation (1) by equation (2) = (−1) · (−1)3 = 1. by equation (4) In a moment we shall see a more efficient algorithm which does not require us to factor integers.

X ← a · x (mod p). while b = 1 (mod p) do m Find the smallest m such that b2 = 1 (mod p). r−m−1 t ← y2 (mod p). 2 y ← t (mod p). r ← m. x ← x · t (mod p). b ← b · y (mod p). return x. where a = 2e · a1 and a1 is odd. We also have the identities, for n odd, 1 n = 1, 2 n = (−1)(n 2 −1)/8 , −1 n = (−1)(n−1)/2 . This now gives us a fast algorithm, which does not require factoring of integers, to determine the Jacobi symbol, and so the Legendre symbol in the case where the denominator is prime. The only factoring required is to extract the even part of a number.

Compute a factorbase F of all prime numbers p less than B. • Find a large number of values of x and y such that x and y are B-smooth and x=y (mod N ). These are called relations on the factorbase. • Using linear algebra modulo 2, find a combination of the relations to give an X and Y with X2 = Y 2 (mod N ). • Attempt to factor N by computing gcd(X − Y, N ). The trick in all algorithms of this form is how to find the relations. All the other details of the algorithms are basically the same. Such a strategy can be used to solve discrete logarithm problems as well, which we shall discuss in Chapter 3.

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Cryptography Made Simple by Nigel Smart

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