Consider an RSA key set with p = 17 , q = 23, N = 23 and e = 3 (as in Figure 1.9). What value of d should be used for the secret key? What is the encryption of the message M = 41 ?

RSA cryptosystem is an asymmetric cryptography algorithm that contains a public as well as a private key. With the use of both the keys, data get more secured and for decryption, the public can be visible to everyone but the private key is shared with the authorized user secretly.

We have, p = 17 , q = 23 , n = 391 and e = 3.

We need to calculate,

$$\begin{gathered} (\mathrm{p}=1)\times (\mathrm{q}-1)=(17-1)\times (23-1) \\ =16\times 22 \\ =352 \end{gathered}$$

To find the inverse of e m od 352 , calculate GCD ( 3,352 ).

As, $352=\left(3\times 117,1\right)$, then

GCD ( 3,352 ) = 1

So, if GCD of any two number equals to one then we can do inverse of the given number.

So, inverse of 3 m od 352

$$\begin{gathered} \mathrm{e}\times \mathrm{d}=1\mathrm{mod}352 \\ 3\times \mathrm{d}=1\mathrm{mod}352 \\ \mathrm{d}=-117 \end{gathered}$$

Therefore, the value of d is -117.

It is given that, $\mathrm{M}=41$. Then,

$$\begin{gathered} \mathrm{E}\left(\mathrm{M}\right)={\mathrm{M}}^{\mathrm{e}}\mathrm{mod}\mathrm{N} \\ ={41}^3 \\ =117\times 41 \\ =105\mathrm{mod}391 \end{gathered}$$

So, Encryption message for M = 41 is 105 mod 391.