135 research outputs found
Explicit CM-theory for level 2-structures on abelian surfaces
For a complex abelian variety with endomorphism ring isomorphic to the
maximal order in a quartic CM-field , the Igusa invariants generate an abelian extension of the reflex field of . In
this paper we give an explicit description of the Galois action of the class
group of this reflex field on . We give a geometric
description which can be expressed by maps between various Siegel modular
varieties. We can explicitly compute this action for ideals of small norm, and
this allows us to improve the CRT method for computing Igusa class polynomials.
Furthermore, we find cycles in isogeny graphs for abelian surfaces, thereby
implying that the `isogeny volcano' algorithm to compute endomorphism rings of
ordinary elliptic curves over finite fields does not have a straightforward
generalization to computing endomorphism rings of abelian surfaces over finite
fields
Modular polynomials for genus 2
Modular polynomials are an important tool in many algorithms involving
elliptic curves. In this article we investigate their generalization to the
genus 2 case following pioneering work by Gaudry and Dupont. We prove various
properties of these genus 2 modular polynomials and give an improved way to
explicitly compute them
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
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
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
Counting Points on Genus 2 Curves with Real Multiplication
We present an accelerated Schoof-type point-counting algorithm for curves of
genus 2 equipped with an efficiently computable real multiplication
endomorphism. Our new algorithm reduces the complexity of genus 2 point
counting over a finite field (\F_{q}) of large characteristic from
(\widetilde{O}(\log^8 q)) to (\widetilde{O}(\log^5 q)). Using our algorithm we
compute a 256-bit prime-order Jacobian, suitable for cryptographic
applications, and also the order of a 1024-bit Jacobian
On polarised class groups of orders in quartic CM-fields
We give an explicit necessary condition for pairs of orders in a quartic
CM-field to have the same polarised class group. This generalises a simpler
result for imaginary quadratic fields. We give an application of our results to
computing endomorphism rings of abelian surfaces over finite fields, and we use
our results to extend a completeness result of Murabayashi and Umegaki to a
list of abelian surfaces over the rationals with complex multiplication by
arbitrary orders.Comment: 19 pages, v2 strengthened results slightly and changed theorem
numbering, v3 further strengthened results and added more details, v4 eased
the presentation but changed notations and numbering, v5 updated references,
v6 removes mistaken "transitivity" statemen
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