4,990 research outputs found
Pairing the Volcano
Isogeny volcanoes are graphs whose vertices are elliptic curves and whose
edges are -isogenies. Algorithms allowing to travel on these graphs were
developed by Kohel in his thesis (1996) and later on, by Fouquet and Morain
(2001). However, up to now, no method was known, to predict, before taking a
step on the volcano, the direction of this step. Hence, in Kohel's and
Fouquet-Morain algorithms, many steps are taken before choosing the right
direction. In particular, ascending or horizontal isogenies are usually found
using a trial-and-error approach. In this paper, we propose an alternative
method that efficiently finds all points of order such that the
subgroup generated by is the kernel of an horizontal or an ascending
isogeny. In many cases, our method is faster than previous methods. This is an
extended version of a paper published in the proceedings of ANTS 2010. In
addition, we treat the case of 2-isogeny volcanoes and we derive from the group
structure of the curve and the pairing a new invariant of the endomorphism
class of an elliptic curve. Our benchmarks show that the resulting algorithm
for endomorphism ring computation is faster than Kohel's method for computing
the -adic valuation of the conductor of the endomorphism ring for small
Tate-Shafarevich groups of constant elliptic curves and isogeny volcanos
We describe the structure of Tate-Shafarevich groups of a constant elliptic
curves over function fields by exploiting the volcano structure of isogeny
graphs of elliptic curves over finite fields
Computing Hilbert class polynomials with the Chinese Remainder Theorem
We present a space-efficient algorithm to compute the Hilbert class
polynomial H_D(X) modulo a positive integer P, based on an explicit form of the
Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the
algorithm uses O(|D|^(1/2+o(1))log P) space and has an expected running time of
O(|D|^(1+o(1)). We describe practical optimizations that allow us to handle
larger discriminants than other methods, with |D| as large as 10^13 and h(D) up
to 10^6. We apply these results to construct pairing-friendly elliptic curves
of prime order, using the CM method.Comment: 37 pages, corrected a typo that misstated the heuristic complexit
Static and Dynamical Susceptibility of LaO1-xFxFeAs
The mechanism of superconductivity and magnetism and their possible interplay
have recently been under debate in pnictides. A likely pairing mechanism
includes an important role of spin fluctuations and can be expressed in terms
of the magnetic susceptibility chi. The latter is therefore a key quantity in
the determination of both the magnetic properties of the system in the normal
state, and of the contribution of spin fluctuations to the pairing potential. A
basic ingredient to obtain chi is the independent-electron susceptibility chi0.
Using LaO1-xFxFeAs as a prototype material, in this report we present a
detailed ab-initio study of chi0(q,omega), as a function of doping and of the
internal atomic positions. The resulting static chi0(q,0) is consistent with
both the observed M-point related magnetic stripe phase in the parent compound,
and with the existence of incommensurate magnetic structures predicted by
ab-initio calculations upon doping.Comment: 15 pages, 8 figure
Isogeny graphs of ordinary abelian varieties
Fix a prime number . Graphs of isogenies of degree a power of
are well-understood for elliptic curves, but not for higher-dimensional abelian
varieties. We study the case of absolutely simple ordinary abelian varieties
over a finite field. We analyse graphs of so-called -isogenies,
resolving that they are (almost) volcanoes in any dimension. Specializing to
the case of principally polarizable abelian surfaces, we then exploit this
structure to describe graphs of a particular class of isogenies known as
-isogenies: those whose kernels are maximal isotropic subgroups
of the -torsion for the Weil pairing. We use these two results to write
an algorithm giving a path of computable isogenies from an arbitrary absolutely
simple ordinary abelian surface towards one with maximal endomorphism ring,
which has immediate consequences for the CM-method in genus 2, for computing
explicit isogenies, and for the random self-reducibility of the discrete
logarithm problem in genus 2 cryptography.Comment: 36 pages, 4 figure
Hard isogeny problems over RSA moduli and groups with infeasible inversion
We initiate the study of computational problems on elliptic curve isogeny
graphs defined over RSA moduli. We conjecture that several variants of the
neighbor-search problem over these graphs are hard, and provide a comprehensive
list of cryptanalytic attempts on these problems. Moreover, based on the
hardness of these problems, we provide a construction of groups with infeasible
inversion, where the underlying groups are the ideal class groups of imaginary
quadratic orders.
Recall that in a group with infeasible inversion, computing the inverse of a
group element is required to be hard, while performing the group operation is
easy. Motivated by the potential cryptographic application of building a
directed transitive signature scheme, the search for a group with infeasible
inversion was initiated in the theses of Hohenberger and Molnar (2003). Later
it was also shown to provide a broadcast encryption scheme by Irrer et al.
(2004). However, to date the only case of a group with infeasible inversion is
implied by the much stronger primitive of self-bilinear map constructed by
Yamakawa et al. (2014) based on the hardness of factoring and
indistinguishability obfuscation (iO). Our construction gives a candidate
without using iO.Comment: Significant revision of the article previously titled "A Candidate
Group with Infeasible Inversion" (arXiv:1810.00022v1). Cleared up the
constructions by giving toy examples, added "The Parallelogram Attack" (Sec
5.3.2). 54 pages, 8 figure
Formation, production and viability of oospores of Phytophthora infestans from potato and Solanum demissum in the Toluca Valley, central Mexico
Aspects of the ecology of oospores of Phytophthora infestans were studied in the highlands of central Mexico. From an investigation of a random sample of strains, it was found that isolates differed in their average capability to form oospores when engaged in compatible pairings. Most crosses produced large numbers of oospores but a few yielded none and some yielded only a few oospores. The results reveal that oospore production and fecundity is dependent on both isolates and the combining ability of a specific combination of parental strains. On average, 14% of the oospores produced were viable as determined by the plasmolysis method. Viability ranged from a low 1% in one cross to a high of 29% in another cross. Oospores were found in 10-20% of naturally infected Solanum demissum leaves from two different collections, and leaflets with two lesions per leaflet produced more oospores than did leaflets with 3-5 lesions per leaflet. There was no consistent trend for preferential mating between isolates from the same location or host
Isogeny graphs with maximal real multiplication
An isogeny graph is a graph whose vertices are principally polarized abelian
varieties and whose edges are isogenies between these varieties. In his thesis,
Kohel described the structure of isogeny graphs for elliptic curves and showed
that one may compute the endomorphism ring of an elliptic curve defined over a
finite field by using a depth first search algorithm in the graph. In dimension
2, the structure of isogeny graphs is less understood and existing algorithms
for computing endomorphism rings are very expensive. Our setting considers
genus 2 jacobians with complex multiplication, with the assumptions that the
real multiplication subring is maximal and has class number one. We fully
describe the isogeny graphs in that case. Over finite fields, we derive a depth
first search algorithm for computing endomorphism rings locally at prime
numbers, if the real multiplication is maximal. To the best of our knowledge,
this is the first DFS-based algorithm in genus 2
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