Chapter 12, Problem 48CRP

Explanation of Solution

Determine the encrypted message 110 with use of the public key:

The public key is $n=91$ and $e=5$. And the message is 110 that is $m=110$.

• To find the binary representation of message value that is $m$, multiply value at once place with 2 raised with power 0 at tens place with 2 raised to the power 1 and at hundred’s place with 2 raised to the power 2.

  $\begin{array}{c}110=1\times 2^2+1\times 2^1+0\times 2^0\\ =1\times 4+1\times 2+0\times 1\\ =4+2+0\\ =6\end{array}$

• Thus, the binary representation of the message is $m=6$.
• The encryption formula for the RSA public-key encryption is,

  $c=m^e\left(\mathrm{mod}n\right)$

  Where,

  Here, $c$ is the encryption key.

Substitute 6 for $m$, 5 for $e$ and 91 for $n$ in the above equation.

$\begin{array}{c}c=6^5\left(\mathrm{mod}91\right)\\ =7776\left(\mathrm{mod}91\right)\\ =41\end{array}$

• To represent the above result in encoded message, convert the result in binary