Which functions can be used as Pseudo Random functions? Can block Ciphers be used for this purpose with our own encryption function?
I would recommend against using ONLY 15-bits as a key for your encryption algorithm. A key size that small can be broken via brute force by any standard machine. There's a reason that AES (where the key can be 128 bits, 192 bits, or 256 bits) or SHA-256 are so widely used. The extended key lengths of those algorithms add security against brute force attacks from adversaries.