Abstract: The invention is a method, system, computer program, computer program article of manufacture, and business method for providing improvements in key generation and cryptographic applications in public key cryptography, by both reducing: 1) the bit-length of public keys and other messages, thereby reducing the bandwidth requirements of telecommunications devices, such as wireless telephone sets, and 2) the computational effort required to generate keys, to encrypt/decrypt and to generate/verify digital signatures. The method of the invention determines a public key having a reduced length and a number p, using GF(p2) arithmetic to achieve GF(p6) security, without explicitly constructing GF(p6).