Math Problem Statement
Alice wants to send a plaintext message π = 13 to Bob secretly using RSA public key cryptosystem. Bob selects π = 13, and π = 11 with π = 13. You have to perform following tasks: a. Compute and list Bobβs public and private keys. b. Compute the ciphertext that Alice will send to Bob using plaintext message π = 13. c. Recover the actual plaintext from the ciphertext sent by Alice.
Solution
Let's walk through the steps to solve the problem using RSA public key cryptosystem.
a. Compute Bobβs public and private keys
-
Choose the prime numbers:
-
Compute (the product of and ):
-
Compute (Eulerβs totient function):
-
Public exponent :
- Given , and it needs to satisfy .
- Since , 13 is valid.
-
Private key :
- To find , we need the modular inverse of modulo , i.e., .
Using the Extended Euclidean Algorithm, we solve for : The solution is (since ).
So, Bob's private key is .
- Bob's public key:
- Bob's private key:
b. Compute the ciphertext that Alice will send to Bob using plaintext
-
Plaintext:
-
Ciphertext is calculated using the formula : We can compute step by step using modular exponentiation:
Then: First, . Now compute: So, the ciphertext is .
c. Recover the actual plaintext from the ciphertext sent by Alice
To decrypt the ciphertext, we use Bob's private key and compute: That is: We will use modular exponentiation to compute this efficiently:
Now we can combine these powers of 65: First, compute , and then:
So, the recovered plaintext is .
Summary of Results:
- Public key:
- Private key:
- Ciphertext:
- Recovered plaintext:
Do you need further details or have any questions?
Here are 5 related questions to extend your understanding:
- What would happen if ?
- How would the encryption process differ with larger prime numbers for and ?
- Can RSA encryption be applied to larger messages, and if so, how?
- How does the Extended Euclidean Algorithm find the modular inverse in RSA?
- What are potential weaknesses in the RSA algorithm, and how can they be mitigated?
Tip: In RSA, choosing large prime numbers makes the encryption more secure, as factoring large numbers becomes more difficult.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Number Theory
Modular Arithmetic
RSA Cryptography
Prime Numbers
Euler's Totient Function
Extended Euclidean Algorithm
Formulas
n = p * q
Ο(n) = (p - 1)(q - 1)
c β‘ m^e mod n
m β‘ c^d mod n
d β‘ e^(-1) mod Ο(n)
Theorems
Euler's Totient Function
Modular Inversion
RSA Algorithm
Suitable Grade Level
University Level (Cryptography Course)