Participants in a cryptosystem select their own elliptic curve and finite field, rather than using a centrally chosen elliptic curve. The curve is chosen from a predetermined set of elliptic curves expressed as Weierstra.beta. model equations. The public key is based on a participant's unique ID, which must be exchanged during communication setup for non-cryptographic reasons, and a randomly chosen bitstring having a length based on security considerations. The public key can be readily constructed from parameters and mapping functions which are known system-wide and from a small amount of participant dependent data.