Authentication and Key Agreement using password provide convenience and amenity, but what human can remember has extremely low entropy. To overcome its defects, AMP(Authentiration and key agreement via Memorable Password) which performs authentication and key agreement securely via low entropy password are presented. AMP uses Diffie-Hellman problem that depends on discrete logarithm problem. Otherwise, this thesis applies elliptic curve cryptosystem to AMP for further efficiency That is, this thesis presents EC-AMP(Elliptic Curve-AMP) protocol based on elliptic curve discrete logarithm problem instead of discrete logarithm problem, and shows its high performance through the implementation. EC-AMP secures against various attacks in the random oracle model just as AMP Thus, we nay supply EC-AMP to the network environment that requires authentication and key agreement to get both convenience and security from elliptic curve discrete logarithm problem.