### 1.3. Public-key Cryptography

In this section, we give a short introduction to the realization of public-key cryptosystems. More specifically, we list some of the computationally intensive mathematical problems and describe how the (apparent) intractability of these problems can be used for designing key pairs. We use some mathematical terms that we will introduce later in this book.

#### 1.3.1. The Mathematical Problems

The security of the public-key cryptosystems is based on the presumed difficulty of solving certain mathematical problems.

##### The integer factorization problem (IFP)

Given the product n = pq of two distinct prime integers p and q, find p and q.

##### The discrete logarithm problem (DLP)

Let G be a finite cyclic (multiplicatively written) ...