Consider the primes p = 5 and q = 11. Let e = 3. (a) Compute pq and (p − 1)(q − 1). Prove that gcd(e, (p − 1)(q − 1)) = 1. Compute the smallest positive integer d such that de = 1 (mod (p − 1)(q − 1)) Equivalently, 3d = 1 (mod 40)
Consider the primes p = 5 and q = 11. Let e = 3. (a) Compute pq and (p − 1)(q − 1). Prove that gcd(e, (p − 1)(q − 1)) = 1. Compute the smallest positive integer d such that de = 1 (mod (p − 1)(q − 1)) Equivalently, 3d = 1 (mod 40)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### RSA Algorithm Exercise
Consider the primes \( p = 5 \) and \( q = 11 \). Let \( e = 3 \).
(a) Here are the steps to follow:
1. **Compute \( pq \) and \( (p - 1)(q - 1) \).**
\[
pq = 5 \times 11 = 55
\]
\[
(p - 1)(q - 1) = (5 - 1)(11 - 1) = 4 \times 10 = 40
\]
2. **Prove that \( \gcd(e, (p - 1)(q - 1)) = 1 \).**
- The greatest common divisor (gcd) is computed as follows:
\[
\gcd(3, 40) = 1
\]
Since 3 and 40 are coprime (they have no common divisors other than 1), \( \gcd(3, 40) = 1 \) holds true.
3. **Compute the smallest positive integer \( d \) such that \[ de \equiv 1 \pmod{(p - 1)(q - 1)} \]**
- This means we need to find \( d \) such that:
\[
de \equiv 1 \pmod{40}
\]
- Equivalently,
\[
3d \equiv 1 \pmod{40}
\]
- The goal is to find the multiplicative inverse of 3 mod 40:
\[
3d \equiv 1 \pmod{40}
\]
- By using the extended Euclidean algorithm or trial and error, \( d \) can be found. In this case,
\[
d = 27
\]
because
\[
(3 \times 27) \equiv 81 \equiv 1 \pmod{40}
\]
Thus, the smallest positive integer \( d \) that satisfies these conditions is \( d = 27 \).
These steps illustrate a simplified example of the principles underlying the RSA encryption algorithm.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa332eac-d846-4704-9340-0a50b86bfcea%2F8036a066-224a-417e-b1ec-11a1cfa107dc%2Fvi6u84_processed.png&w=3840&q=75)
Transcribed Image Text:### RSA Algorithm Exercise
Consider the primes \( p = 5 \) and \( q = 11 \). Let \( e = 3 \).
(a) Here are the steps to follow:
1. **Compute \( pq \) and \( (p - 1)(q - 1) \).**
\[
pq = 5 \times 11 = 55
\]
\[
(p - 1)(q - 1) = (5 - 1)(11 - 1) = 4 \times 10 = 40
\]
2. **Prove that \( \gcd(e, (p - 1)(q - 1)) = 1 \).**
- The greatest common divisor (gcd) is computed as follows:
\[
\gcd(3, 40) = 1
\]
Since 3 and 40 are coprime (they have no common divisors other than 1), \( \gcd(3, 40) = 1 \) holds true.
3. **Compute the smallest positive integer \( d \) such that \[ de \equiv 1 \pmod{(p - 1)(q - 1)} \]**
- This means we need to find \( d \) such that:
\[
de \equiv 1 \pmod{40}
\]
- Equivalently,
\[
3d \equiv 1 \pmod{40}
\]
- The goal is to find the multiplicative inverse of 3 mod 40:
\[
3d \equiv 1 \pmod{40}
\]
- By using the extended Euclidean algorithm or trial and error, \( d \) can be found. In this case,
\[
d = 27
\]
because
\[
(3 \times 27) \equiv 81 \equiv 1 \pmod{40}
\]
Thus, the smallest positive integer \( d \) that satisfies these conditions is \( d = 27 \).
These steps illustrate a simplified example of the principles underlying the RSA encryption algorithm.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 27 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
