The El-Gamal AA_β Public Key Cryptosystem is a new asymmetric cryptosystem based on the piecewise AA_β-function. The AA_β-function which is essentially a one way Boolean function was motivated by the squaring and multiplying process while computing g^a (mod p) in the Diffie Hellman key exchange procedure and also computing C \equiv M^e (mod N) and M \equiv C^d (mod N) in the RSA cryptosystem. It was also motivated by the add and double point operation E=kG in the elliptic curve cryptosystem. The hard mathematical problem surrounding this newly designed asymmetric cryptosystem is the NP-complete problem known as the subset sum problem. The El-Gamal AA_β Public Key Cryptosystem mimics the El-Gamal Cryptosystem and the Elliptic Curve Cryptosystem by sending a two parameter ciphertext to the recipient.