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View article: PEARL-SCALLOP: Parameter Extension Applicable in Real Life for SCALLOP
PEARL-SCALLOP: Parameter Extension Applicable in Real Life for SCALLOP Open
A crucial ingredient for many cryptographic primitives such as key exchange protocols and advanced signature schemes is a commutative group action where the structure of the underlying group can be computed efficiently. SCALLOP provides su…
View article: Vignéras orbifolds: isospectrality, regulators, and torsion homology
Vignéras orbifolds: isospectrality, regulators, and torsion homology Open
We develop a new approach to the isospectrality of the orbifolds constructed by Vignéras. We give fine sufficient criteria for i-isospectrality in given degree i and for representation equivalence. These allow us to produce very small exot…
View article: The supersingular Endomorphism Ring and One Endomorphism problems are equivalent
The supersingular Endomorphism Ring and One Endomorphism problems are equivalent Open
The supersingular Endomorphism Ring problem is the following: given a supersingular elliptic curve, compute all of its endomorphisms. The presumed hardness of this problem is foundational for isogeny-based cryptography. The One Endomorphis…
View article: Computing groups of Hecke characters
Computing groups of Hecke characters Open
We describe algorithms to represent and compute groups of Hecke characters. We make use of an id{è}lic point of view and obtain the whole family of such characters, including transcendental ones. We also show how to isolate the algebraic c…
View article: Norm relations and computational problems in number fields
Norm relations and computational problems in number fields Open
For a finite group $G$, we introduce a generalization of norm relations in the group algebra $\mathbb Q[G]$. We give necessary and sufficient criteria for the existence of such relations and apply them to obtain relations between the arith…
View article: Computing fundamental domains for arithmetic Kleinian groups
Computing fundamental domains for arithmetic Kleinian groups Open
We present an algorithm which, given an arithmetic Kleinian group Γ, returns a fundamental domain and a finite presentation for Γ with a computable isomorphism.
View article: Sorting and labelling integral ideals in a number field
Sorting and labelling integral ideals in a number field Open
We define a scheme for labelling and ordering integral ideals of number fields, including prime ideals as a special case. The order we define depends only on the choice of a monic irreducible integral defining polynomial for each field $K$…
View article: Aurel Page - Construction of subfields and abelian overfields
Aurel Page - Construction of subfields and abelian overfields Open
This tutorial:
- construction of subfields of a number field
- construction of abelian extensions of a number field
These are old functionalities but we made a number of changes to them. If you want to record the commands we will type d…
View article: Computing isogenies from modular equations in genus two
Computing isogenies from modular equations in genus two Open
We present an algorithm solving the following problem: given two genus 2 curves over a field k with isogenous Jacobians, compute such an isogeny explicitly. This isogeny can be either an l-isogeny or, in the real multiplication case, an is…
View article: Computing isogenies from modular equations between Jacobians of genus 2 curves
Computing isogenies from modular equations between Jacobians of genus 2 curves Open
Let k be a field of large enough characteristic. We present an algorithm solving the following problem: given two genus 2 curves over k with isogenous Jacobians, compute an isogeny between them explicitly. This isogeny can be either an ℓ-i…
View article: Group representations in the homology of 3-manifolds
Group representations in the homology of 3-manifolds Open
If M is a manifold with an action of a group G , then the homology group H_1(M,\mathbb Q) is naturally a \mathbb Q[G] -module, where \mathbb Q[G] denotes the rational group ring. We prove that for every finite group G , and for every \math…
View article: Torsion homology and regulators of isospectral manifolds
Torsion homology and regulators of isospectral manifolds Open
Given a finite group G, a G-covering of closed Riemannian manifolds, and a so-called G-relation, a construction of Sunada produces a pair of manifolds M_1 and M_2 that are strongly isospectral. Such manifolds have the same dimension and th…
View article: Computing arithmetic Kleinian groups
Computing arithmetic Kleinian groups Open
Arithmetic Kleinian groups are arithmetic lattices in PSL_2(C). We present an algorithm which, given such a group Gamma, returns a fundamental domain and a finite presentation for Gamma with a computable isomorphism.