For one-dimensional infrastructures, we have a circle together with a finite, non-empty set and an injective map . Now , where and is the one-dimensional lattice . Hence, one could say that an -dimensional infrastructure is a torus together with and as above. From the discussion in the remarks of this post we see that we need some kind of reduction map to define giant steps (and also -representations) in the one-dimensional case, even though there a pretty canonical reduction map is given. In the case of , we do not have something similar to a given positive direction. Moreover, definiting the “nearest” element of a finite subset of to some is even more complicated and offers more choices which appear more or less obvious. Only the selection of different norms on lead to several possible definitions of such a map. Hence, we should require such a map in the definition:
Let be a lattice. Then, an -dimensional infrastructure is a non-empty finite set together with an injective map and another map satisfying .
Again, as in the one-dimensional case, one can define giant steps:
Moreover, one gets the same relation between reduction maps and -representations, whence we define
Then the map
is a bijection, and we can use this bijection to equip with a group law by
We say that an -dimensional infrastructure with lattice is discrete if , and if does not depends on fractions. To make the last part more precise, define
if , induces a map . Now, that does not depends on fractions simply means that factors through , i.e. that we can write with .
Moreover, if in the following we specify discrete infrastructures, we often just define for values in . In that case, for elements , define .
In case is discrete, consider the subset
is an isomorphism.
Finite Abelian Groups as Infrastructures.
Let be a finite abelian group, generated by . Consider the relation lattice for , defined by
Then is the kernel of , , and
is a group isomorphism. Define , and (or, more precisely, ); then is an -dimensional infrastructure. Moreover, for , we have , whence equals the group operation of . Hence, every finite abelian group can be seen in a natural way as an infrastructure.
Moreover, this shows that can be thought of as an analogue to the discrete logarithm map, and is an analogue of the power map . In particular, we obtained the goal described in the first post of this series: we found a generalization of the discrete logarithm problem to a non-associative algebraic structure. In the next post, I will how such infrastructures can be obtained from global fields; this gives a rich source of examples for -dimensional infrastructures.
What about baby steps?
Note that in the above discussion, I simply ignored baby steps. In the one-dimensional case, has a canonical direction (namely the positive one) and so has , whence saying “go to the next element” makes sense. Opposed to that, in , there are infinitely many directions, no one better than another. Even if we fix a direction, “go to the next element in that direction” seems to not really make sense. So far, I have not seen any definition of baby steps in this case which works for all -dimensional infrastructures.
Note that in the case of infrastructures obtained from global fields, one has some kind of canonical baby steps (even though there are still some choices left). In fact, there are of them. To define them, though, one needs more information than just , and : one needs information about a -st dimension, both for constructing the reduction map and for defining baby steps.