Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
### 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 ciphertext To 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): \[
Transcribed Image Text:### 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 ciphertext To 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): \[
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS