31 research outputs found
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
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
Class number approximation in cubic function fields
We develop explicitly computable bounds for the order of the
Jacobian of a cubic function field. We use approximations via
truncated Euler products and thus derive effective methods
of computing the order of the Jacobian of a cubic function field.
Also, a detailed discussion of the zeta function of a cubic
function field extension is included
Explicit infrastructure for real quadratic function fields and real hyperelliptic curves
In 1989, Koblitz first proposed the Jacobian of a an imaginary hyperelliptic curve for use in public-key cryptographic protocols. This concept is a generalization of elliptic curve cryptography. It can be used with the same assumed key-per-bit strength for small genus. More recently, real hyperelliptic curves of small genus have been introduced as another source for cryptographic protocols. The arithmetic is more involved than its imaginary counterparts and it is based on the so-called infrastructure of the set of reduced principal ideals in the ring of regular functions of the curve. This infrastructure is an interesting phenomenon. The main purpose of this article is to explain the infrastructure in explicit terms and thus extend Shanks\u27 infrastructure ideas in real quadratic number fields to the case of real quadratic congruence function fields and their curves. Hereby, we first present an elementary introduction to the continued fraction expansion of real quadratic irrationalities and then generalize important results for reduced ideals
Picard curves over Q with good reduction away from 3
Inspired by methods of N. P. Smart, we describe an algorithm to determine all
Picard curves over Q with good reduction away from 3, up to Q-isomorphism. A
correspondence between the isomorphism classes of such curves and certain
quintic binary forms possessing a rational linear factor is established. An
exhaustive list of integral models is determined, and an application to a
question of Ihara is discussed.Comment: 27 pages; A previous lemma was incorrect and has been removed;
Corrected computation has identified 18 new such curves (63 in total
Explicit Methods in Number Theory
These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes the Sato-Tate conjecure, Langlands programme, function fields, L-functions and many other topics