The ElGamal Cryptosystem: 1. Alice chooses a large finite cyclic group G and a generator g E G. She chooses a random integer a E 10,1,2,..., IG-1), which she keeps secret. She computes A- g. Her public key is (G, g, A) 2. Bob wants to send Alice a message mEG. He obtains a copy of her public key and chooses a randon b E {0, 1, 2, . . . , IG-1), then computes B-gb and c-Abm. The ciphertext is the pair (B, c) E G G. He sends this to Alice 3. To decrytp Bob's message, Alice computes (Bam 1. (i) What common elements do the ElGamal cryptosystem and Diffie-Hellmarn (ii) What are the encryption and decryption functions in the ElGamal Cryp- (iii) Is it necessary for the element g chosen by Alice to be a generator in G? (iv) The Discrete Logarithm Problem is believed to be hard (that is, no efficient key exchange share? tosystem? Prove that encryption and decryption are inverse operations. Why/why not? algorithm currently exists). Explain how the ElGamal Cryptosystem uses this to ensure key security. The ElGamal Cryptosystem: 1. Alice chooses a large finite cyclic group G and a generator g E G. She chooses a random integer a E 10,1,2,..., IG-1), which she keeps secret. She computes A- g. Her public key is (G, g, A) 2. Bob wants to send Alice a message mEG. He obtains a copy of her public key and chooses a randon b E {0, 1, 2, . . . , IG-1), then computes B-gb and c-Abm. The ciphertext is the pair (B, c) E G G. He sends this to Alice 3. To decrytp Bob's message, Alice computes (Bam 1. (i) What common elements do the ElGamal cryptosystem and Diffie-Hellmarn (ii) What are the encryption and decryption functions in the ElGamal Cryp- (iii) Is it necessary for the element g chosen by Alice to be a generator in G? (iv) The Discrete Logarithm Problem is believed to be hard (that is, no efficient key exchange share? tosystem? Prove that encryption and decryption are inverse operations. Why/why not? algorithm currently exists). Explain how the ElGamal Cryptosystem uses this to ensure key security