In the Diffie-Hellman key exchange protocol with public parameters (g = 3, p = 31), suppose Alice's private key is a = 5 and she receives Bob's public key B = 11. What is the shared key K between Alice and Bob?

Question

BlackTom AI · Accepted Answer

To determine the shared key K in Diffie-Hellman with the given parameters, we use the standard formula K = B^a mod p, where B is Bob's public key, a is Alice's private key, and p is the prime modulus.
Option a (12): If we test 11^5 mod 31, we get 6, not 12. The value 12 would require a different exponent or a different base; it does not align with the modular exponentiation in this setup. 
Option b (22): This would imply a different result from 11^5 mod 31, but carrying out the modular multiplication step by step (11^2 ≡ 28, 11^4 ≡ 9, 11^5 ≡ 9·11 ≡ 99 ≡ 6 mod 31) shows 22 is incorrect.
Option c (2): This is another common accidental small residue, but the actual computation 11^5 mod 31 yields 6, so 2 is not correct here. 
Option d (6): This matches the correct modular exponentiation: compute sequentially or use exponentiation by squaring. 11^2 ≡ 28 (mod 31), 11^4 ≡ 28^2 ≡ 784 ≡ 9 (mod 31), and 11^5 ≡ 11^4·11 ≡ 9·11 ≡ 99 ≡ 6 (mod 31). Thus the shared key K = 6.
In summary, the correct shared key arises from evaluating B^a mod p with B = 11, a = 5, p = 31, yielding 6, while each of the other options does not satisfy the modular exponentiation result in this configuration.

In the Diffie-Hellman key exchange protocol with public parameters (g = 3, p = 31), suppose Alice's private key is a = 5 and she receives Bob's public key B = 11. What is the shared key K between Alice and Bob?单项选择题

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