  The Discrete Logarithm Problem is well-known among cryptographers, for its
computational hardness that grants security to some of the most commonly used
cryptosystems these days. Still, many of these are limited to a small number of
candidate algebraic structures which permit implementing the algorithms. In
order to extend the applicability of discrete-logarithm-based cryptosystems to
a much richer class of algebraic structures, we present a generalized form of
exponential function. Our extension relaxes some assumptions on the exponent,
which is no longer required to be an integer. Using an axiomatic
characterization of the exponential function, we show how to construct mappings
that obey the same rules as exponentials, but can raise vectors to the power of
other vectors in an algebraically sound manner. At the same time, computational
hardness is not affected (in fact, the problem could possibly be strengthened).
Setting up standard cryptosystems in terms of our generalized exponential
function is simple and requires no change to the existing security proofs. This
opens the field for building much more general schemes than the ones known so
far.
