By Nigel Smart

In this introductory textbook the writer explains the major issues in cryptography. he is taking a contemporary method, the place defining what's intended through "secure" is as vital as growing anything that achieves that aim, and safeguard definitions are important to the dialogue throughout.

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J ← 1. if a < 0 then a ← −a. if b (mod 4) = 3 then j ← −j. while a = 0 do while a (mod 2) = 0 do a ← a/2. if b (mod 8) = 3 or b (mod 8) = 5 then j ← −j. (a, b) ← (b, a). if a (mod 4) = 3 and b (mod 4) = 3 then j ← −j. a ← a (mod b). if b = 1 then return j. return 0. • n is the product of two primes, n = p · q: • Qn ⊂ Jn . • #Qn = #(Jn \ Qn ) = (p − 1)(q − 1)/4. The sets Qn and Jn will be seen to be important in a number of algorithms and protocols, especially in the case where n is a product of two primes.

Let B be an integer. An integer N is called B-smooth if every prime factor p of N is less than B. For example N = 278 · 389 · 113 is 12-smooth. Sometimes we say that the number is just smooth if the bound B is small compared with N . The number of y-smooth numbers which are less than x is given by the function ψ(x, y). This is a rather complicated function which is approximated by ψ(x, y) ≈ xρ(u) where ρ is the Dickman–de Bruijn function and log x . log y The Dickman–de Bruijn function ρ is deﬁned as the function which satisﬁes the following diﬀerentialdelay equation u · ρ (u) + ρ(u − 1) = 0, for u > 1.

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. In this section, we explain the parts of the modern factoring algorithms which are common and justify why they work. One way of looking at such algorithms is in the context of computational group theory. The factorbase is essentially a set of generators of the group (Z/N Z)∗ , whilst the relations are relations between the generators of this group.