17,105 research outputs found
Average-Case Complexity
We survey the average-case complexity of problems in NP.
We discuss various notions of good-on-average algorithms, and present
completeness results due to Impagliazzo and Levin. Such completeness results
establish the fact that if a certain specific (but somewhat artificial) NP
problem is easy-on-average with respect to the uniform distribution, then all
problems in NP are easy-on-average with respect to all samplable distributions.
Applying the theory to natural distributional problems remain an outstanding
open question. We review some natural distributional problems whose
average-case complexity is of particular interest and that do not yet fit into
this theory.
A major open question whether the existence of hard-on-average problems in NP
can be based on the PNP assumption or on related worst-case assumptions.
We review negative results showing that certain proof techniques cannot prove
such a result. While the relation between worst-case and average-case
complexity for general NP problems remains open, there has been progress in
understanding the relation between different ``degrees'' of average-case
complexity. We discuss some of these ``hardness amplification'' results
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
Computational Indistinguishability between Quantum States and Its Cryptographic Application
We introduce a computational problem of distinguishing between two specific
quantum states as a new cryptographic problem to design a quantum cryptographic
scheme that is "secure" against any polynomial-time quantum adversary. Our
problem, QSCDff, is to distinguish between two types of random coset states
with a hidden permutation over the symmetric group of finite degree. This
naturally generalizes the commonly-used distinction problem between two
probability distributions in computational cryptography. As our major
contribution, we show that QSCDff has three properties of cryptographic
interest: (i) QSCDff has a trapdoor; (ii) the average-case hardness of QSCDff
coincides with its worst-case hardness; and (iii) QSCDff is computationally at
least as hard as the graph automorphism problem in the worst case. These
cryptographic properties enable us to construct a quantum public-key
cryptosystem, which is likely to withstand any chosen plaintext attack of a
polynomial-time quantum adversary. We further discuss a generalization of
QSCDff, called QSCDcyc, and introduce a multi-bit encryption scheme that relies
on similar cryptographic properties of QSCDcyc.Comment: 24 pages, 2 figures. We improved presentation, and added more detail
proofs and follow-up of recent wor
Cryptography from tensor problems
We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler
Constructing lattice points for numerical integration by a reduced fast successive coordinate search algorithm
In this paper, we study an efficient algorithm for constructing node sets of
high-quality quasi-Monte Carlo integration rules for weighted Korobov, Walsh,
and Sobolev spaces. The algorithm presented is a reduced fast successive
coordinate search (SCS) algorithm, which is adapted to situations where the
weights in the function space show a sufficiently fast decay. The new SCS
algorithm is designed to work for the construction of lattice points, and, in a
modified version, for polynomial lattice points, and the corresponding
integration rules can be used to treat functions in different kinds of function
spaces. We show that the integration rules constructed by our algorithms
satisfy error bounds of optimal convergence order. Furthermore, we give details
on efficient implementation such that we obtain a considerable speed-up of
previously known SCS algorithms. This improvement is illustrated by numerical
results. The speed-up obtained by our results may be of particular interest in
the context of QMC for PDEs with random coefficients, where both the dimension
and the required numberof points are usually very large. Furthermore, our main
theorems yield previously unknown generalizations of earlier results.Comment: 33 pages, 2 figure
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