Final answer:
The primary purpose of the public key (g, q, g, h) in the public-key encryption scheme is for the sender to encrypt a message, specifically a plaintext bit b, for secure communication.
Step-by-step explanation:
The primary purpose of the public key (g, q, g, h) in the described public-key encryption scheme is for encryption by the sender. In public-key cryptography, which includes the ElGamal encryption scheme, the public key is used to encrypt messages while the private key is retained by the receiver to decrypt those messages. The public key components, in this case, consist of the base g, the prime number q, and the value h which is a transformation of the base g using the private key x. The sender uses the public key to transform a plaintext bit b, into a ciphertext that the receiver can later decrypt using their private key. The public key components (g, q, h) are used by the sender to generate the ciphertext during the encryption process. The sender uses these values along with some random values to produce the ciphertext that is then sent to the receiver. The receiver, who possesses the private key, can then decrypt the ciphertext to obtain the original message.