Question

In an RSA cryptosystem, p = 7and q = 11(as in Figure 1.9). Find appropriate exponents and .

Short answer

The correct exponent of d is 37 and e is 13.

Step-by-step solution

Introduction

RSA cryptosystem are asymmetric cryptography algorithm in which it contains public as well as private key. By the use of both the keys data get more secured and for decryption public can be visible to everyone but private key is shared with the authorized user secretly.

Find the value of  

We have given that, p = 7 , q = 11 , n =$\mathrm{p}\times \mathrm{q},$

Then

$$\begin{gathered} \mathrm{n}=7\times 11 \\ =77 \end{gathered}$$

Now, find $\varphi \left(\mathrm{n}\right)=(\mathrm{p}-1)\times (\mathrm{q}-1)$.

$$\begin{gathered} \varphi \left(n\right)=(p-1)\times (q-1) \\ =(7-1)\times (11-1) \\ =6\times 10 \\ =60 \end{gathered}$$

Let’s, assume e as private key, such that

$(\mathrm{e}\times \mathrm{d})\mathrm{modphi}\left(\mathrm{n}\right)=1$

The value of e is 13 as it is the next prime number after 11

So,

$$\begin{gathered} (13\times \mathrm{d})\mathrm{mod}60=1 \\ \mathrm{d}=37 \end{gathered}$$

The correct exponent of d is 37 and e is 13.