Optimization of the arithmetic of the ideal class group for genus 4 hyperelliptic curves over projective coordinates
Rodrigo Abarzúa Nicolas Thériault Roberto Avanzi Ismael Soto Miguel Alfaro
Advances in Mathematics of Communications 2010, 4(2): 115-139 doi: 10.3934/amc.2010.4.115
The aim of this paper is to reduce the number of operations in Cantor's algorithm for the Jacobian group of hyperelliptic curves for genus 4 in projective coordinates. Specifically, we developed explicit doubling and addition formulas for genus 4 hyperelliptic curves over binary fields with $h(x)=1$. For these curves, we can perform a divisor doubling in $63M+19S$, while the explicit adding formula requires $203M+18S,$ and the mixed coordinates addition (in which one point is given in affine coordinates) is performed in $165M+15S$.
  These formulas can be useful for public key encryption in some environments where computing the inverse of a field element has a high computational cost (either in time, power consumption or hardware price), in particular with embedded microprocessors.
keywords: projective coordinates. explicit formulas genus 4 Hyperelliptic curves binary field
A filtering method for the hyperelliptic curve index calculus and its analysis
Roberto Avanzi Nicolas Thériault
Advances in Mathematics of Communications 2010, 4(2): 189-213 doi: 10.3934/amc.2010.4.189
We describe a filtering technique improving the performance of index-calculus algorithms for hyperelliptic curves. Filtering is a stage taking place between the relation search and the linear algebra. Its purpose is to eliminate redundant or duplicate relations, as well as reducing the size of the matrix, thus decreasing the time required for the linear algebra step.
  This technique, which we call harvesting, is in fact a new strategy that subtly alters the whole index calculus algorithm. In particular, it changes the relation search to find many times more relations than variables, after which a selection process is applied to the set of the relations - the harvesting process. The aim of this new process is to extract a (slightly) overdetermined submatrix which is as small as possible. Furthermore, the size of the factor base also has to be readjusted, in order to keep the (extended) relation search faster than it would have been in an index calculus algorithm without harvesting. The size of the factor base must also be chosen to guarantee that the final matrix will be indeed smaller than it would be in an optimised index calculus without harvesting, thus also speeding up the linear algebra step.
  The version of harvesting presented here is an improvement over an earlier version by the same authors. By means of a new selection algorithm, time-complexity can be reduced from quadratic to linear (in the size of the input), thus making its running time effectively negligible with respect to the rest of the index calculus algorithm. At the same time we make the process of harvesting more effective - in the sense that the final matrix should (on average) be smaller than with the earlier approach.
  We present an analysis of the impact of harvesting (for instance, we show that its usage can improve index calculus performance by more than 30% in some cases), we show that the impact on matrix size is essentially independent on the genus of the curve considered, and provide an heuristic argument in support of the effectiveness of harvesting as one parameter (which defines how far the relation search is pushed) increases.
keywords: Jacobians Hyperelliptic curves harvesting. index calculus filtering
Efficient halving for genus 3 curves over binary fields
Peter Birkner Nicolas Thériault
Advances in Mathematics of Communications 2010, 4(1): 23-47 doi: 10.3934/amc.2010.4.23
In this article, we deal with fast arithmetic in the Picard group of hyperelliptic curves of genus 3 over binary fields. We investigate both the optimal performance curves, where $h(x)=1$, and the more general curves where the degree of $h(x)$ is 1, 2 or 3. For the optimal performance curves, we provide explicit halving and doubling formulas; not only for the most frequent case but also for all possible special cases that may occur when performing arithmetic on the proposed curves. In this situation, we show that halving offers equivalent performance to that of doubling when computing scalar multiples (by means of an halve-and-add algorithm) in the divisor class group.
    For the other types of curves where halving may give performance gains (when the group order is twice an odd number), we give explicit halving formulas which outperform the corresponding doubling formulas by about 10 to 20 field multiplications per halving. These savings more than justify the use of halvings for these curves, making them significantly more efficient than previously thought. For halving on genus 3 curves there is no previous work published so far.
keywords: binary field doubling halving divisor class explicit formulas genus 3 Hyperelliptic curve cryptography.
Trisection for supersingular genus $2$ curves in characteristic $2$
Josep M. Miret Jordi Pujolàs Nicolas Thériault
Advances in Mathematics of Communications 2014, 8(4): 375-387 doi: 10.3934/amc.2014.8.375
By reversing reduction in divisor class arithmetic we provide efficient trisection algorithms for supersingular Jacobians of genus $2$ curves over finite fields of characteristic $2$. With our technique we obtain new results for these Jacobians: we show how to find their $3$-torsion subgroup, we prove there is none with $3$-torsion subgroup of rank $3$ and we prove that the maximal $3$-power order subgroup is isomorphic to either $\mathbb{Z}/3^{v}\mathbb{Z}$ or $(\mathbb{Z}/3^{\frac v2}\mathbb{Z})^2$ or $(\mathbb{Z}/3^{\frac v4}\mathbb{Z})^4$, where $v$ is the $3$-adic valuation $v_{3}$(#Jac(C)$(\mathbb{F}_{2^m})$). Ours are the first trisection formulae available in literature.
keywords: Hyperelliptic curve explicit formulas. binary field trisection supersingular divisor class genus 2

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