Answer: Therefore, the shared key between Party A and Party B is 4.
Step-by-step explanation:
For Abdullah and Aymen:
Given:
P = 23 (modulus)
G = 5 (primitive root)
a = 6 (Abdullah's private key)
b = 15 (Aymen's private key)
a. To find Abdullah's public key:
Abdullah's public key = G^a (mod P)
Abdullah's public key = 5^6 (mod 23)
Abdullah's public key = 15625 (mod 23)
Abdullah's public key = 16
Therefore, Abdullah's public key is 16.
b. To find Aymen's public key:
Aymen's public key = G^b (mod P)
Aymen's public key = 5^15 (mod 23)
Aymen's public key = 30517578125 (mod 23)
Aymen's public key = 8
Therefore, Aymen's public key is 8.
c. To find the shared secret key between Abdullah and Aymen:
Shared secret key = (Abdullah's public key)^b (mod P) = (Aymen's public key)^a (mod P)
Shared secret key = 16^15 (mod 23) = 8^6 (mod 23)
Shared secret key = 1073741824 (mod 23) = 16
Therefore, the shared secret key between Abdullah and Aymen is 16.
For Party A and Party B:
Given:
P = 7 (modulus)
G = 3 (primitive root)
Party A chooses a = 2 (Party A's private key)
Party B chooses b = 5 (Party B's private key)
a. To find Party A's public key:
Party A's public key = G^a (mod P)
Party A's public key = 3^2 (mod 7)
Party A's public key = 9 (mod 7)
Party A's public key = 2
Therefore, Party A's public key is 2.
b. To find Party B's public key:
Party B's public key = G^b (mod P)
Party B's public key = 3^5 (mod 7)
Party B's public key = 243 (mod 7)
Party B's public key = 5
Therefore, Party B's public key is 5.
c. To find the shared key between Party A and Party B:
Shared key = (Party B's public key)^a (mod P) = (Party A's public key)^b (mod P)
Shared key = 5^2 (mod 7) = 2^5 (mod 7)
Shared key = 25 (mod 7) = 4