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Abstract Algebra: An Introduction
Found in: Page 437
Abstract Algebra: An Introduction

Abstract Algebra: An Introduction

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

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Short Answer

Find the decoding algorithm for the code in Exercise 3.

The required algorithm for decoding the code Cidmod 2773=Bi , for the received message in the block form C1C2C3C4...Cm, are given by B1B2B3B4...Bm.

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Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

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=3 in the RSA algorithm.

Here,

n=2773=77×59

Let, p=77 , q=59 and

k=p−1q−1=77−159−1=7658=2668

Compute the value of d using 3d≡1mod 2663 .

Thus, d=888.

Now, compute the values Cidmod 2773=Bi, for the received message in the block form, C1C2C3C4...Cm.

The decoded messages are given by B1B2B3B4...Bm.

Thus is the required algorithm.

Most popular questions for Math Textbooks

Prove this version of Fermat’s Little Theorem: If p is a prime and a∈ℤ , then ap=a(mod p) [Hint: Consider two case, p|a and p|a; use Lemma 13.2 in the second case. ]

Use a calculator and the RSA encoding algorithm with e=3 , n=2773 to encode these messages:

(a) GO HOME (b) COME BACK (c) DROP DEAD

[Hint: Use 2-letter blocks and don't omit spaces.]

Question: Let p be a prime and k,a∈ℤ such that p≁a and 0 < k < p . Prove thatka≢0modp . [Hint: Theorem 1.5]

Let C be the coded form of a message that was encoded by using the RSA algorithm. Suppose that you discover that C and the encoding modulus n are not relatively prime. Explain how you could factor n and thus find the decoding algorithm. [The probability of such a C occurring is less than 10-99 when the prime factors p, q, of n have more than 100 digits.]
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