(7) (RSA public key cryptosystem) Bob chooses the secret primes p = 11 and q = 23. Bob chooses the encryp- tion exponent e = 7. Bob publishes N = 253 and e = 7. Alice chooses the plaintext m₁ = 5. Compute the ciphertext c₁ that Alice sends to Bob.
(7) (RSA public key cryptosystem) Bob chooses the secret primes p = 11 and q = 23. Bob chooses the encryp- tion exponent e = 7. Bob publishes N = 253 and e = 7. Alice chooses the plaintext m₁ = 5. Compute the ciphertext c₁ that Alice sends to Bob.
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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![### RSA Public Key Cryptosystem Example
**Problem:**
Bob chooses the secret primes \( p = 11 \) and \( q = 23 \). Bob chooses the encryption exponent \( e = 7 \). Bob publishes \( N = 253 \) and \( e = 7 \).
Alice chooses the plaintext \( m_1 = 5 \). Compute the ciphertext \( c_1 \) that Alice sends to Bob.
**Solution:**
To encrypt the plaintext \( m_1 = 5 \) using Bob's public key values \( N = 253 \) and \( e = 7 \), Alice uses the RSA encryption formula:
\[ c \equiv m^e \, (\text{mod} \, N) \]
Given:
- \( m = 5 \)
- \( e = 7 \)
- \( N = 253 \)
1. Compute \( 5^7 \):
\[
5^7 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 78125
\]
2. Compute \( 78125 \mod 253 \):
\[
78125 \div 253 \approx 308.8745 \quad \Rightarrow \quad 78125 = 253 \times 308 + 181 \quad \Rightarrow \quad 78125 \mod 253 = 181
\]
Therefore, the ciphertext \( c_1 \) that Alice sends to Bob is:
\[ c_1 = 181 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa332eac-d846-4704-9340-0a50b86bfcea%2F931cd6b1-9d6b-40b4-b333-04129f5a69e7%2Ftr1rpfo_processed.png&w=3840&q=75)
Transcribed Image Text:### RSA Public Key Cryptosystem Example
**Problem:**
Bob chooses the secret primes \( p = 11 \) and \( q = 23 \). Bob chooses the encryption exponent \( e = 7 \). Bob publishes \( N = 253 \) and \( e = 7 \).
Alice chooses the plaintext \( m_1 = 5 \). Compute the ciphertext \( c_1 \) that Alice sends to Bob.
**Solution:**
To encrypt the plaintext \( m_1 = 5 \) using Bob's public key values \( N = 253 \) and \( e = 7 \), Alice uses the RSA encryption formula:
\[ c \equiv m^e \, (\text{mod} \, N) \]
Given:
- \( m = 5 \)
- \( e = 7 \)
- \( N = 253 \)
1. Compute \( 5^7 \):
\[
5^7 = 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 = 78125
\]
2. Compute \( 78125 \mod 253 \):
\[
78125 \div 253 \approx 308.8745 \quad \Rightarrow \quad 78125 = 253 \times 308 + 181 \quad \Rightarrow \quad 78125 \mod 253 = 181
\]
Therefore, the ciphertext \( c_1 \) that Alice sends to Bob is:
\[ c_1 = 181 \]
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