25 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
The arithmetic of Jacobian groups of superelliptic cubics
International audienceWe present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity. The first algorithm, inspired by Cantor's reduction for hyperelliptic curves, is easily implemented with a few lines of code, making use of a polynomial arithmetic package. We prove explicit reducedness criteria for superelliptic curves of genus 3 and 4, which show the correctness of the algorithm. The second approach, quite general in nature and applicable to further classes of curves, uses the FGLM algorithm for switching between Gröbner bases for different orderings. Carrying out the computations symbolically, we obtain explicit reduction formulae in terms of the input data
A double large prime variation for small genus hyperelliptic index calculus
International audienceIn this article, we examine how the index calculus approach for computing discrete logarithms in small genus hyperelliptic curves can be improved by introducing a double large prime variation. Two algorithms are presented. The first algorithm is a rather natural adaptation of the double large prime variation to the intended context. On heuristic and experimental grounds, it seems to perform quite well but lacks a complete and precise analysis. Our second algorithm is a considerably simplified variant, which can be analyzed easily. The resulting complexity improves on the fastest known algorithms. Computer experiments show that for hyperelliptic curves of genus three, our first algorithm surpasses Pollard's Rho method even for rather small field sizes
Hyperelliptic Curve Cryptosystems: Closing the Performance Gap to Elliptic Curves (Update)
For most of the time since they were proposed, it was widely
believed that hyperelliptic curve cryptosystems (HECC) carry a
substantial performance penalty compared to elliptic curve
cryptosystems (ECC) and are, thus, not too attractive for
practical applications. Only quite recently improvements have been
made, mainly restricted to curves of genus 2. The work at hand
advances the state-of-the-art considerably in several aspects.
First, we generalize and improve the closed formulae for the group
operation of genus 3 for HEC defined over fields of characteristic
two. For certain curves we achieve over 50% complexity improvement
compared to the best previously published results. Second, we
introduce a new complexity metric for ECC and HECC defined over
characteristic two fields which allow performance comparisons of
practical relevance. It can be shown that the HECC performance is
in the range of the performance of an ECC; for specific
parameters HECC can even possess a lower complexity than an ECC at
the same security level. Third, we describe the first
implementation of a HEC cryptosystem on an embedded (ARM7)
processor. Since HEC are particularly attractive for constrained
environments, such a case study should be of relevance
A New Method for Decomposition in the Jacobian of Small Genus Hyperelliptic Curves
Decomposing a divisor over a suitable factor basis in the Jacobian of a hyperelliptic curve is a crucial step in an
index calculus algorithm for the discrete log problem in the Jacobian. For small genus curves, in the year 2000, Gaudry had proposed
a suitable factor basis and a decomposition method. In this work, we provide a new method for decomposition over the same factor
basis. The advantage of the new method is that it admits a sieving technique which removes smoothness checking of polynomials
required in Gaudry\u27s method. Also, the total number of additions in the Jacobian required by the new method is less than
that required by Gaudry\u27s method. The new method itself is quite simple and we present some example decompositions and timing
results of our implementation of the method using Magma