Groups from Cyclic Infrastructures and Pohlig-Hellman in Certain Infrastructures

In discrete logarithm based cryptography, a method by Pohlig and Hellman allows solving the discrete logarithm problem efficiently if the group order is known and has no large prime factors. The consequence is that such groups are avoided. In the past, there have been proposals for cryptography based on cyclic infrastructures. We will show that the Pohlig-Hellman method can be adapted to certain cyclic infrastructures, which similarly implies that certain infrastructures should not be used for cryptography. This generalizes a result by M\"uller, Vanstone and Zuccherato for infrastructures obtained from hyperelliptic function fields. We recall the Pohlig-Hellman method, define the concept of a cyclic infrastructure and briefly describe how to obtain such infrastructures from certain function fields of unit rank one. Then, we describe how to obtain cyclic groups from discrete cyclic infrastructures and how to apply the Pohlig-Hellman method to compute absolute distances, which is in general a computationally hard problem for cyclic infrastructures. Moreover, we give an algorithm which allows to test whether an infrastructure satisfies certain requirements needed for applying the Pohlig-Hellman method, and discuss whether the Pohlig-Hellman method is applicable in infrastructures obtained from number fields. Finally, we discuss how this influences cryptography based on cyclic infrastructures.Comment: 14 page

The Infrastructure of a Global Field of Arbitrary Unit Rank

In this paper, we show a general way to interpret the infrastructure of a global field of arbitrary unit rank. This interpretation generalizes the prior concepts of the giant step operation and f-representations, and makes it possible to relate the infrastructure to the (Arakelov) divisor class group of the global field. In the case of global function fields, we present results that establish that effective implementation of the presented methods is indeed possible, and we show how Shanks' baby-step giant-step method can be generalized to this situation.Comment: Revised version. Accepted for publication in Math. Com