I begin with giving an abstract definition of a one-dimensional infrastructure.

Definition (One-dimensional infrastructure).

Let be a real number. A one-dimensional infrastructure of circumference is a pair , where is a finite set and is an injective map.

If you interpret as a circle of circumference (think of it as folding up the real line, such that two numbers whose difference is an integer multiple of are identified), a one-dimensional infrastructure can be seen as a circle with a finite number of dots on it. The map gives the distance between and some element on the circle, whence is called the distance map.

Now one can define two operations on a one-dimensional infrastructure. Due to Shanks’ nomenclature, these are called baby steps and giant steps. To define a baby step, let . Then consider the set . It is non-empty as and bounded from below. Moreover, it is discrete as is finite; therefore, exists and , say with . In that case, we define . This gives a bijective map which, in case , has no fixed points. If is interpreted as a circle and identified with , will send each point to the “next one” in positive direction on the circle.

To define giant steps, let . For that, note that is naturally a group, whence we can add and . Now , but in general . But we can use a similar trick as in the baby step case: we jump back to the previous point of . For that, define . It is bounded from above, non-empty and discrete, whence exists with , say for ; then we define . This gives a binary operation which is in general not associative.

But even though, we have in general, assuming that is small (here, we identify with its smallest non-negative representant). More precisely, we have with , whence the giant step operation is “almost” associative.

Finite Cyclic Groups as One-dimensional Infrastructures.

Let be a finite cyclic group of order . For , one can write with ; note that is the discrete logarithm of with respect to . Hence, we get the isomorphism induced by . As , we get the injective map , , turning into a one-dimensional infrastructure.

Let ; then we get and , i.e. baby steps are multiplications by the generator and the giant steps equals the group operation. In particular, this provides an example for giant steps being associative.

Therefore, one-dimensional infrastructures can be seen as generalizations of finite cyclic groups.

Remarks.

Finally, we want to sketch some ideas, which will allow generalizing infrastructures to higher dimensions. For that, let be a one-dimensional infrastructure. First, define the map as follows. For , define . Again, is non-empty, bounded from below and discrete, whence exists and , say for some . Define . Then satisfies and for all .

If would be any other map satisfying , one would obtain another giant step function , . In case comes from a finite cyclic group, as above, would again be the group operation. If this is not the case, and could be two distinct binary operations on . If satisfies for all , we would also have

This shows that our choice of is rather random; we could also define with , or chose such that , with some additional condition to rule out ties. Any other arbitrary choice of is also possible, as long as is satisfied. We will later see that our definition of is exactly the one we obtain in a canonical way if we obtain infrastructures from global fields of unit rank one. We call such maps reduction maps.

On the other side, the inverse question, given and , how to find , is in general hard. In fact, a result by Victor Shoup says that for a black box group, there exists no deterministic algorithm which can find the in time less than steps in any case, provided the prime is the order of . In other words: the Discrete Logarithm Problem (DLP) is hard in general: given with , find with .

Consider the subgroup generated by . It is of the form , and the map , is an epimorphism. Let with ; then induces an isomorphism . The inverse map , is called the discrete logarithm with base , denoted by . Hence, the DLP is the problem of computing the function .

In 1976, Whitfield Diffie and Martin Hellman came up with a key exchange protocol based on the fact that, given , and , one probably needs to solve at least one DLP (to obtain or ) to be able to compute . And nine years later, Taher ElGamal presented an encryption scheme which is also based on the DLP.

First, these systems were originally used with or , the multiplicative groups of finite (prime) fields, but can in fact be used with any group. In 1985, Neal Koblitz and Victor Miller independently proposed to use elliptic curves over finite fields , which yield abelian groups of size . Later, groups obtained from hyperelliptic curves or curves were proposed.

There have also been proposals to replace groups by weaker structures, for example semigroups or certain loops. In general, one can use any algebraic structure which allows to define for , such that can be computed efficiently. As for that, one usually wants , we merely require the operation to be power associative.

Now one can ask if it is possible to drop this requirement and replace exponentiation and discrete logarithms by something similar. So far, the only such algebraic structure which has a huge class of instances which can be easily obtained from algebraic objects and in which one can efficiently compute are infrastructures. In the following articles I first want to explain and discuss the definition of infrastructures and show how finite abelian groups can be interpreted as infrastructures. Then I want to explain how infrastructures can be obtained from global fields.

In case you want to read up more details, please consult the preprint of my paper The Infrastructure of a Global Field of Arbitrary Unit Rank or my PhD thesis (click Volltext).