200 research outputs found
Discrete logarithms in curves over finite fields
A survey on algorithms for computing discrete logarithms in Jacobians of
curves over finite fields
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
A Computational Introduction to Elliptic and Hyperelliptic Curve Cryptography
At its core, cryptography relies on problems that are simple to construct but difficult to solve unless certain information (the “key”) is known. Many of these problems come from number theory and group theory. One method of obtaining groups from which to build cryptosystems is to define algebraic curves over finite fields and then derive a group structure from the set of points on those curves. This thesis serves as an exposition of Elliptic Curve Cryptography (ECC), preceded by a discussion of some basic cryptographic concepts and followed by a glance into one generalization of ECC: cryptosystems based on hyperelliptic curves
Curves, Jacobians, and Cryptography
The main purpose of this paper is to give an overview over the theory of
abelian varieties, with main focus on Jacobian varieties of curves reaching
from well-known results till to latest developments and their usage in
cryptography. In the first part we provide the necessary mathematical
background on abelian varieties, their torsion points, Honda-Tate theory,
Galois representations, with emphasis on Jacobian varieties and hyperelliptic
Jacobians. In the second part we focus on applications of abelian varieties on
cryptography and treating separately, elliptic curve cryptography, genus 2 and
3 cryptography, including Diffie-Hellman Key Exchange, index calculus in Picard
groups, isogenies of Jacobians via correspondences and applications to discrete
logarithms. Several open problems and new directions are suggested.Comment: 66 page
Quantum algorithms for problems in number theory, algebraic geometry, and group theory
Quantum computers can execute algorithms that sometimes dramatically
outperform classical computation. Undoubtedly the best-known example of this is
Shor's discovery of an efficient quantum algorithm for factoring integers,
whereas the same problem appears to be intractable on classical computers.
Understanding what other computational problems can be solved significantly
faster using quantum algorithms is one of the major challenges in the theory of
quantum computation, and such algorithms motivate the formidable task of
building a large-scale quantum computer. This article will review the current
state of quantum algorithms, focusing on algorithms for problems with an
algebraic flavor that achieve an apparent superpolynomial speedup over
classical computation.Comment: 20 pages, lecture notes for 2010 Summer School on Diversities in
Quantum Computation/Information at Kinki Universit
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