In the Diffie-Hellman key exchange protocol with (g = 3, p = 31), suppose Alice's private key is a = 7. What is Alice's public key A?

Question

BlackTom AI · Accepted Answer

We need to compute Alice's public key A using the Diffie-Hellman formula A = g^a mod p with g = 3, p = 31, and a = 7.
First, determine powers of 3 modulo 31 to reach the exponent 7 in manageable steps.
Option a (A = 10): Check whether 3^7 mod 31 equals 10. We calculate 3^7 mod 31 step by step to verify; it does not equal 10, so this option is incorrect.
Option b (A = 25): Similarly, test whether 3^7 mod 31 equals 25. Since the actual result is not 25, this option is not correct.
Option c (A = 5): Again, evaluate 3^7 mod 31 to see if it is 5. The modular reduction yields a different value, so this choice is incorrect.
Option d (A = 17): Compute 3^7 mod 31 explicitly: 3^2 = 9, 3^3 = 27, 3^4 = 27*3 = 81 ≡ 81 − 62 = 19 (mod 31), 3^5 ≡ 19*3 = 57 ≡ 57 − 31 = 26, 3^6 ≡ 26*3 = 78 ≡ 78 − 62 = 16, 3^7 ≡ 16*3 = 48 ≡ 48 − 31 = 17. This shows the correct public key is 17, matching option d.
In short, the correct public key is 3^7 mod 31 = 17, while the other options do not satisfy the modular exponentiation result and hence are incorrect.

In the Diffie-Hellman key exchange protocol with (g = 3, p = 31), suppose Alice's private key is a = 7. What is Alice's public key A?单项选择题

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