The goal of this research is to solve some important problems in nnnntorus-based, pairing-based, multilinear, and elliptic curve and abelian variety cryptography. Elliptic curve cryptography helps to secure the Internet and is used by the U.S. and other governments and institutions to provide secure communication. Abelian variety cryptography includes elliptic and hyperelliptic curve cryptography, scales well to high security levels, and is especially advantageous in constrained environments.