Applications to cryptography of twisting commutative algebraic groups

We give an overview on twisting commutative algebraic groups and applications to discrete log-based cryptography. We explain how discrete log-based cryptography over extension fields can be reduced to cryptography in primitive subgroups. Primitive subgroups in turn are part of a general theory of tensor products of commutative algebraic groups and Galois modules (or twists of commutative algebraic groups), and this underlying mathematical theory can be used to shed light on discrete log-based cryptosystems. We give a number of concrete examples, to illustrate the definitions and results in an explicit way.