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question:Find the shared secret key between two parties using Diffie-Hellman key exchange with the following information: - The prime number used is p = 17 - The primitive root of p is g = 3 - The private key of the first party is a = 5 - The private key of the second party is b = 11
answer:To find the shared secret key using the Diffie-Hellman key exchange, we need to follow these steps: 1. Calculate the public key of the first party (A): A = g^a mod p 2. Calculate the public key of the second party (B): B = g^b mod p 3. Calculate the shared secret key using the public and private keys: - First party calculates: S = B^a mod p - Second party calculates: S = A^b mod p Let's calculate the public keys of both parties: 1. A = 3^5 mod 17 = 243 mod 17 = 15 2. B = 3^11 mod 17 = 177147 mod 17 = 6 Now, let's calculate the shared secret key: - First party: S = 6^5 mod 17 = 7776 mod 17 = 10 - Second party: S = 15^11 mod 17 = 437893890380859375 mod 17 = 10 The shared secret key between the two parties is 10.
question:Alice and Bob agree to use a prime number p = 17 and a base g = 3 for their key exchange. Alice chooses the secret integer a = 5, and sends Bob A = g^a mod p. Bob chooses the secret integer b = 10, and sends Alice B=g^b mod p. What is the shared secret key that both Alice and Bob calculate using their secret integer and received integer?
answer:First, let's calculate the integers Alice and Bob send to each other: Alice sends Bob A = g^a mod p = 3^5 mod 17 A = 243 mod 17 = 9 Bob sends Alice B = g^b mod p = 3^10 mod 17 B = 59049 mod 17 = 12 Now, Alice and Bob will calculate the shared secret key using their secret integers and received integers: Alice calculates the shared secret key as K_A = B^a mod p = 12^5 mod 17 K_A = 248832 mod 17 = 3 Bob calculates the shared secret key as K_B = A^b mod p = 9^10 mod 17 K_B = 3486784401 mod 17 = 3 Both Alice and Bob have calculated the same shared secret key, K = 3.
question:Alice and Bob are trying to establish a shared secret key using Diffie-Hellman key exchange. Alice chooses a prime number p=17 and primitive root of p g=3. She chooses her secret key a=5 and sends Bob A=3^5 mod 17. Bob chooses his secret key b=3 and computes B=3^3 mod 17. What is the shared secret key that Alice and Bob can use for secure communication?
answer:First, let's compute A and B: A = 3^5 mod 17 = 243 mod 17 = 6 B = 3^3 mod 17 = 27 mod 17 = 10 Now, Alice and Bob can compute the shared secret key using A and B: Alice computes the shared secret key as: K = B^a mod p = 10^5 mod 17 = 100000 mod 17 = 3 Bob computes the shared secret key as: K = A^b mod p = 6^3 mod 17 = 216 mod 17 = 3 The shared secret key that Alice and Bob can use for secure communication is 3.
question:Suppose Alice and Bob are communicating over a public channel and want to exchange a secret key. They agree to use the Diffie-Hellman key exchange protocol with prime number p = 23 and base g = 5. Alice chooses a secret number a = 6, and Bob chooses a secret number b = 9. Calculate the secret key that they will both compute and use for secure communication.
answer:To calculate the secret key using the Diffie-Hellman key exchange protocol, follow these steps: 1. Alice computes A = g^a mod p, where g is the base, a is Alice's secret number, and p is the prime number. 2. Bob computes B = g^b mod p, where b is Bob's secret number. 3. Alice and Bob exchange their computed values A and B over the public channel. 4. Alice computes the secret key as K_A = B^a mod p. 5. Bob computes the secret key as K_B = A^b mod p. 6. If the protocol is successful, K_A and K_B will be the same, and this will be the shared secret key. Now, let's compute the secret key using the given values: 1. Alice computes A = 5^6 mod 23 = 15625 mod 23 = 8. 2. Bob computes B = 5^9 mod 23 = 1953125 mod 23 = 2. 3. Alice and Bob exchange A and B, so Alice has B = 2 and Bob has A = 8. 4. Alice computes K_A = 2^6 mod 23 = 64 mod 23 = 18. 5. Bob computes K_B = 8^9 mod 23 = 134217728 mod 23 = 18. Since K_A and K_B are the same, the shared secret key is 18. Alice and Bob will use this secret key for secure communication.