### RSA Encryption Example Problem and Solution#### Problem Statement:Bob’s RSA public key has a modulus \( N = 12191 \) and exponent \( e = 37 \). Alice sends Bob the ciphertext \( c = 587 \). Unfortunately, Bob has chosen too small a modulus. Help Eve by factoring \( N \) and decrypting Alice’s message. (Hint: \( N \) has a factor smaller than 100.)#### Explanation:In this problem, we are dealing with RSA encryption, where:- \( N \) is the modulus- \( e \) is the public exponent- \( c \) is the ciphertextTo decrypt the message, you would generally need the private key. The private key can be derived if you can factorize \( N \) into its prime factors. Given the hint that \( N \) has a factor smaller than 100, we need to find these factors to proceed with the decryption process.#### Steps to Solve:1. **Factorize \( N \)**: Start by finding the prime factors of \( 12191 \). As the hint suggests, look for factors smaller than 100.2. **Calculate \( \phi(N) \)**: Once you have the factors, compute Euler's totient function \( \phi(N) \).3. **Find the private exponent \( d \)**: Compute \( d \) using the relation \( d \cdot e \equiv 1 \,(\text{mod}\, \phi(N)) \).4. **Decrypt the ciphertext \( c \)**: Calculate \( m \), the original message, using the relation \( m = c^d \,(\text{mod}\, N) \).By following the above steps, you will be able to decrypt the message that Alice sent to Bob.#### Detailed Steps:**1. Factorize \( N \)**: - Test divisibility by prime numbers less than 100 to find that \( N = 12191 \). - Notice that \( 12191 \div 11 = 1109 \), so one of the factors is 11. - Continue factoring \( 1109 \) and find that it divides by 101 (as \( 1109 = 11 \times 101 \)).**2. Calculate \( \phi(N) \)**: - Using the prime factors (11 and 101): \[

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3.7. Bob's RSA public key has modulus N = 12191 and exponent e = 37. Alice sends Bob the ciphertext c= 587. Unfortunately, Bob has chosen too small a modu- lus. Help Eve by factoring N and decrypting Alice's message. (Hint. N has a factor smaller than 100.)

3.7. Bob's RSA public key has modulus N = 12191 and exponent e = 37. Alice sends Bob the ciphertext c= 587. Unfortunately, Bob has chosen too small a modu- lus. Help Eve by factoring N and decrypting Alice's message. (Hint. N has a factor smaller than 100.)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.7: Introduction To Coding Theory (optional)

Problem 12E: Suppose that the check digit is computed as described in Example . Prove that transposition errors...

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