778 research outputs found
Finding hidden Borel subgroups of the general linear group
We present a quantum algorithm for solving the hidden subgroup problem in the
general linear group over a finite field where the hidden subgroup is promised
to be a conjugate of the group of the invertible lower triangular matrices. The
complexity of the algorithm is polynomial when size of the base field is not
much smaller than the degree.Comment: 12pt, 10 page
E_11 and M Theory
We argue that eleven dimensional supergravity can be described by a
non-linear realisation based on the group E_{11}. This requires a formulation
of eleven dimensional supergravity in which the gravitational degrees of
freedom are described by two fields which are related by duality. We show the
existence of such a description of gravity.Comment: 21 pages, some typos corrected and two references adde
Hidden Stabilizers, the Isogeny To Endomorphism Ring Problem and the Cryptanalysis of pSIDH
The Isogeny to Endomorphism Ring Problem (IsERP) asks to compute the
endomorphism ring of the codomain of an isogeny between supersingular curves in
characteristic given only a representation for this isogeny, i.e. some data
and an algorithm to evaluate this isogeny on any torsion point. This problem
plays a central role in isogeny-based cryptography; it underlies the security
of pSIDH protocol (ASIACRYPT 2022) and it is at the heart of the recent attacks
that broke the SIDH key exchange. Prior to this work, no efficient algorithm
was known to solve IsERP for a generic isogeny degree, the hardest case
seemingly when the degree is prime.
In this paper, we introduce a new quantum polynomial-time algorithm to solve
IsERP for isogenies whose degrees are odd and have many prime
factors. As main technical tools, our algorithm uses a quantum algorithm for
computing hidden Borel subgroups, a group action on supersingular isogenies
from EUROCRYPT 2021, various algorithms for the Deuring correspondence and a
new algorithm to lift arbitrary quaternion order elements modulo an odd integer
with many prime factors to powersmooth elements.
As a main consequence for cryptography, we obtain a quantum polynomial-time
key recovery attack on pSIDH. The technical tools we use may also be of
independent interest
Coset Symmetries in Dimensionally Reduced Bosonic String Theory
We discuss the dimensional reduction of various effective actions,
particularly that of the closed Bosonic string and pure gravity, to two and
three dimensions. The result for the closed Bosonic string leads to coset
symmetries which are in agreement with those recently predicted and argued to
be present in a new unreduced formulation of this theory. We also show that
part of the Geroch group appears in the unreduced duality symmetric formulation
of gravity recently proposed. We conjecture that this formulation can be
extended to a non-linear realisation based on a Kac-Moody algebra which we
identify. We also briefly discuss the proposed action of Bosonic M-theory.Comment: Reference adde
Lectures on Gauged Supergravity and Flux Compactifications
The low-energy effective theories describing string compactifications in the
presence of fluxes are so-called gauged supergravities: deformations of the
standard abelian supergravity theories. The deformation parameters can be
identified with the various possible (geometric and non-geometric) flux
components. In these lecture notes we review the construction of gauged
supergravities in a manifestly duality covariant way and illustrate the
construction in several examples.Comment: 48 pages, lectures given at the RTN Winter School on Strings,
Supergravity and Gauge Theories, CERN, January 200
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