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5E

Expert-verified
Found in: Page 437

### Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

# Find the decoding algorithm for the code in Exercise 3.

The required algorithm for decoding the code ${\mathrm{C}}_{\mathrm{i}}^{\mathrm{d}}\left(\mathrm{mod}\text{}2773\right)={\mathrm{B}}_{\mathrm{i}}$ , for the received message in the block form $\begin{array}{cccccc}{\mathrm{C}}_{1}& {\mathrm{C}}_{2}& {\mathrm{C}}_{3}& {\mathrm{C}}_{4}& ...& {\mathrm{C}}_{\mathrm{m}}\end{array}$, are given by $\begin{array}{cccccc}{\mathrm{B}}_{1}& {\mathrm{B}}_{2}& {\mathrm{B}}_{3}& {\mathrm{B}}_{4}& ...& {\mathrm{B}}_{\mathrm{m}}\end{array}$.

See the step by step solution

## Define RSA algorithm.

RSA algorithm is used to encode and decode messages. RSA algorithm is a public key cryptography algorithm. This form of the algorithm is asymmetric cryptographic algorithm and uses two different type of keys.

## Determine the decoding algorithm

Take $n=2773$ and $e\mathbf{=}3$ in the RSA algorithm.

Here,

$\begin{array}{c}n=2773\\ =77×59\end{array}$

Let, $p=77$ , $q=59$ and

$\begin{array}{c}k=\left(p-1\right)\left(q-1\right)\\ =\left(77-1\right)\left(59-1\right)\\ =\left(76\right)\left(58\right)\\ =2668\end{array}$

Compute the value of d using $3d\equiv 1\left(\mathrm{mod}\text{}2663\right)$ .

Thus, $d\mathbf{=}888$.

Now, compute the values ${C}_{i}^{d}\left(\mathrm{mod}\text{}2773\right)={B}_{i}$, for the received message in the block form, $\begin{array}{cccccc}{C}_{1}& {C}_{2}& {C}_{3}& {C}_{4}& ...& {C}_{m}\end{array}$.

The decoded messages are given by $\begin{array}{cccccc}{B}_{1}& {B}_{2}& {B}_{3}& {B}_{4}& ...& {B}_{m}\end{array}$.

Thus is the required algorithm.