This paper discusses elliptic curve cryptosystems as analogs of public key cryptosystems based on the multiplicative group of a finite field. The author argues that elliptic curve cryptosystems may be more secure because the elliptic curve discrete logarithm problem is likely harder than the classical discrete logarithm problem, especially over GF(2^n). The paper explores the structure of elliptic curves over finite fields, their group properties, and how they can be used in cryptographic systems. It describes two elliptic curve analogs of existing cryptosystems: the Massey-Omura system and the ElGamal system. The paper also discusses the importance of choosing elliptic curves with orders that are nonsmooth (i.e., divisible by large primes) to make the discrete logarithm problem more difficult to solve. The author provides examples of elliptic curves and their properties, and discusses the probability that a randomly chosen point generates a large subgroup. The paper also addresses the question of primitive points on elliptic curves and the probability that a point generates the entire group. The author concludes that elliptic curve cryptosystems are a promising area of research and that further study is needed to fully understand their security properties.This paper discusses elliptic curve cryptosystems as analogs of public key cryptosystems based on the multiplicative group of a finite field. The author argues that elliptic curve cryptosystems may be more secure because the elliptic curve discrete logarithm problem is likely harder than the classical discrete logarithm problem, especially over GF(2^n). The paper explores the structure of elliptic curves over finite fields, their group properties, and how they can be used in cryptographic systems. It describes two elliptic curve analogs of existing cryptosystems: the Massey-Omura system and the ElGamal system. The paper also discusses the importance of choosing elliptic curves with orders that are nonsmooth (i.e., divisible by large primes) to make the discrete logarithm problem more difficult to solve. The author provides examples of elliptic curves and their properties, and discusses the probability that a randomly chosen point generates a large subgroup. The paper also addresses the question of primitive points on elliptic curves and the probability that a point generates the entire group. The author concludes that elliptic curve cryptosystems are a promising area of research and that further study is needed to fully understand their security properties.