Math Problem Statement
Solution
The image contains a cryptographic problem about the Diffie-Hellman key exchange. Below is a detailed analysis of the question:
Question (translated and analyzed):
What is the shared secret key, computed using the Diffie-Hellman process? If:
- One party chooses their private key to be
- The other party selects their private key to be
Solution:
The Diffie-Hellman key exchange process involves two parties (often called Alice and Bob) who agree on:
- A prime number (the modulus).
- A generator (base value used for exponentiation).
Each participant chooses a private key:
- Alice chooses .
- Bob chooses .
Then:
- Alice sends Bob the value .
- Bob sends Alice the value .
Finally, both compute the shared secret key independently:
- Alice computes .
- Bob computes . Since both computations result in the same value, they both end up with the same shared secret.
Step-by-Step Calculations:
-
Alice sends Bob:
-
Bob sends Alice:
- First, calculate .
- Now, .
-
Alice computes the shared secret:
- First, calculate .
- Now, .
-
Bob computes the shared secret:
- First, calculate .
- Now, .
Thus, the shared secret key is:
Do you want more details on any step or further explanation on Diffie-Hellman exchange?
Related Questions:
- What is the purpose of the generator in Diffie-Hellman?
- How can you enhance the security of Diffie-Hellman with larger primes?
- What makes the shared secret in Diffie-Hellman secure against eavesdropping?
- Can the Diffie-Hellman method be used for symmetric encryption key sharing?
- What vulnerabilities does Diffie-Hellman have in modern cryptography?
Tip:
In cryptography, it’s important to choose sufficiently large prime numbers to prevent brute force attacks or discrete logarithm attacks.
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Math Problem Analysis
Mathematical Concepts
Cryptography
Modular Arithmetic
Number Theory
Formulas
A = g^a mod n
B = g^b mod n
K = B^a mod n
K = A^b mod n
Theorems
Diffie-Hellman Key Exchange
Suitable Grade Level
Undergraduate - Computer Science or Cryptography course
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