6,187 research outputs found
Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices
Let \orig{A} be any matrix and let be a slight random perturbation of
\orig{A}. We prove that it is unlikely that has large condition number.
Using this result, we prove it is unlikely that has large growth factor
under Gaussian elimination without pivoting. By combining these results, we
bound the smoothed precision needed by Gaussian elimination without pivoting.
Our results improve the average-case analysis of Gaussian elimination without
pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 1997).Comment: corrected some minor mistake
Towards Human Computable Passwords
An interesting challenge for the cryptography community is to design
authentication protocols that are so simple that a human can execute them
without relying on a fully trusted computer. We propose several candidate
authentication protocols for a setting in which the human user can only receive
assistance from a semi-trusted computer --- a computer that stores information
and performs computations correctly but does not provide confidentiality. Our
schemes use a semi-trusted computer to store and display public challenges
. The human user memorizes a random secret mapping
and authenticates by computing responses
to a sequence of public challenges where
is a function that is easy for the
human to evaluate. We prove that any statistical adversary needs to sample
challenge-response pairs to recover , for
a security parameter that depends on two key properties of . To
obtain our results, we apply the general hypercontractivity theorem to lower
bound the statistical dimension of the distribution over challenge-response
pairs induced by and . Our lower bounds apply to arbitrary
functions (not just to functions that are easy for a human to evaluate),
and generalize recent results of Feldman et al. As an application, we propose a
family of human computable password functions in which the user
needs to perform primitive operations (e.g., adding two digits or
remembering ), and we show that .
For these schemes, we prove that forging passwords is equivalent to recovering
the secret mapping. Thus, our human computable password schemes can maintain
strong security guarantees even after an adversary has observed the user login
to many different accounts.Comment: Fixed bug in definition of Q^{f,j} and modified proofs accordingl
The LU-LC conjecture is false
The LU-LC conjecture is an important open problem concerning the structure of
entanglement of states described in the stabilizer formalism. It states that
two local unitary equivalent stabilizer states are also local Clifford
equivalent. If this conjecture were true, the local equivalence of stabilizer
states would be extremely easy to characterize. Unfortunately, however, based
on the recent progress made by Gross and Van den Nest, we find that the
conjecture is false.Comment: Added a new part explaining how the counterexamples are foun
A note on the growth factor in Gaussian elimination for generalized Higham matrices
The Higham matrix is a complex symmetric matrix A=B+iC, where both B and C
are real, symmetric and positive definite and is the
imaginary unit. For any Higham matrix A, Ikramov et al. showed that the growth
factor in Gaussian elimination is less than 3. In this paper, based on the
previous results, a new bound of the growth factor is obtained by using the
maximum of the condition numbers of matrixes B and C for the generalized Higham
matrix A, which strengthens this bound to 2 and proves the Higham's conjecture.Comment: 8 pages, 2 figures; Submitted to MOC on Dec. 22 201
All CHSH polytopes
The correlations that admit a local hidden-variable model are described by a
family of polytopes, whose facets are the Bell inequalities. The CHSH
inequality is the simplest such Bell inequality and is a facet of every Bell
polytope. We investigate for which Bell polytopes the CHSH inequality is also
the unique (non-trivial) facet. We prove that the CHSH inequality is the unique
facet for all bipartite polytopes where at least one party has a binary choice
of dichotomic measurements, irrespective of the number of measurement settings
and outcomes for the other party. Based on numerical results, we conjecture
that it is also the unique facet for all bipartite polytopes involving two
measurements per party where at least one measurement is dichotomic. Finally,
we remark that these two situations can be the only ones for which the CHSH
inequality is the unique facet, i.e., any polytope that does not correspond to
one of these two cases necessarily has facets that are not of the CHSH form. As
a byproduct of our approach, we derive a new family of facet inequalities
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