By Nigel Smart
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.
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Extra info for Cryptography Made Simple
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.
Cryptography Made Simple by Nigel Smart