52,104 research outputs found
A Comment on Gu Map-1
Gu map-1 is a modified version of GGH map. It uses same ideal lattices for constructing the trapdoors, while the novelty is that no encodings of zero are given. In this short paper we show that Gu map-1 cannot be used for the instance of witness encryption (WE) based on the hardness of 3-exact cover problem. That is, if Gu map-1 is used for such instance, we can break it by solving a combined 3-exact cover problem. The reason is just that no encodings of zero are given
Pseudo-Diagonals and Uniqueness Theorems
We examine a certain type of abelian C*-subalgebras that allow one to give a
unified treatment of two uniqueness theorems: for graph C*-algebras and for
certain reduced crossed products
Topological phases with generalized global symmetries
We present simple lattice realizations of symmetry-protected topological
(SPT) phases with -form global symmetries where charged excitations have
spatial dimensions. Specifically, we construct space-dimensional models
supported on a -colorable graph by using a family of unitary phase
gates, known as multi-qubit control- gates in quantum information community.
In our construction, charged excitations of different dimensionality may
coexist and form a short-range entangled state which is protected by symmetry
operators of different dimensionality. Non-triviality of proposed models, in a
sense of quantum circuit complexity, is confirmed by studying protected
boundary modes, gauged models and corresponding gapped domain walls. We also
comment on applications of our construction to quantum error-correcting codes,
and discuss corresponding fault-tolerant logical gates.Comment: 32 pages, 17 figures, single column (v2, corrected minor mistakes and
typos, to appear in PRB
Homomorphisms between diffeomorphism groups
For r at least 3, p at least 2, we classify all actions of the groups
Diff^r_c(R) and Diff^r_+(S1) by C^p -diffeomorphisms on the line and on the
circle. This is the same as describing all nontrivial group homomorphisms
between groups of compactly supported diffeomorphisms on 1- manifolds. We show
that all such actions have an elementary form, which we call topologically
diagonal. As an application, we answer a question of Ghys in the 1-manifold
case: if M is any closed manifold, and Diff(M)_0 injects into the
diffeomorphism group of a 1-manifold, must M be 1 dimensional? We show that the
answer is yes, even under more general conditions. Several lemmas on subgroups
of diffeomorphism groups are of independent interest, including results on
commuting subgroups and flows.Comment: Contains corrections and additional references. A revised version
will appear in Ergodic Theory and Dynamical System
On the Maslov class rigidity for coisotropic submanifolds
We define the Maslov index of a loop tangent to the characteristic foliation
of a coisotropic submanifold as the mean Conley--Zehnder index of a path in the
group of linear symplectic transformations, incorporating the "rotation" of the
tangent space of the leaf -- this is the standard Lagrangian counterpart -- and
the holonomy of the characteristic foliation. Furthermore, we show that, with
this definition, the Maslov class rigidity extends to the class of the
so-called stable coisotropic submanifolds including Lagrangian tori and stable
hypersurfaces.Comment: 18 pages; v2 minor corrections, references update
An introduction to loopoids
We discuss a concept of loopoid as a non-associative generalization of
(Brandt) groupoid. We introduce and study also an interesting class of more
general objects which we call semiloopoids. A differential version of loopoids
is intended as a framework for Lagrangian discrete mechanics.Comment: 9 pages, proceedings of LOOPS'1
On realizing homology classes by maps of restricted complexity
We show that in every codimension greater than one there exists a mod 2
homology class in some closed manifold (of sufficiently high dimension) which
cannot be realized by an immersion of closed manifolds. The proof gives
explicit obstructions (in terms of cohomology operations) for realizability of
mod 2 homology classes by immersions. We also prove the corresponding result in
which the word `immersion' is replaced by `map with some restricted set of
multi-singularities'.Comment: 13 pages; Final version, to appear in Bull. Lond. Math. So
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