45 research outputs found
Improved Low-qubit Hidden Shift Algorithms
Hidden shift problems are relevant to assess the quantum security of various
cryptographic constructs. Multiple quantum subexponential time algorithms have
been proposed. In this paper, we propose some improvements on a polynomial
quantum memory algorithm proposed by Childs, Jao and Soukharev in 2010. We use
subset-sum algorithms to significantly reduce its complexity. We also propose
new tradeoffs between quantum queries, classical time and classical memory to
solve this problem
Exact Algorithms for 0-1 Integer Programs with Linear Equality Constraints
In this paper, we show -time and -space exact
algorithms for 0-1 integer programs where constraints are linear equalities and
coefficients are arbitrary real numbers. Our algorithms are quadratically
faster than exhaustive search and almost quadratically faster than an algorithm
for an inequality version of the problem by Impagliazzo, Lovett, Paturi and
Schneider (arXiv:1401.5512), which motivated our work. Rather than improving
the time and space complexity, we advance to a simple direction as inclusion of
many NP-hard problems in terms of exact exponential algorithms. Specifically,
we extend our algorithms to linear optimization problems
Deterministic Time-Space Tradeoffs for k-SUM
Given a set of numbers, the -SUM problem asks for a subset of numbers
that sums to zero. When the numbers are integers, the time and space complexity
of -SUM is generally studied in the word-RAM model; when the numbers are
reals, the complexity is studied in the real-RAM model, and space is measured
by the number of reals held in memory at any point.
We present a time and space efficient deterministic self-reduction for the
-SUM problem which holds for both models, and has many interesting
consequences. To illustrate:
* -SUM is in deterministic time and space
. In general, any
polylogarithmic-time improvement over quadratic time for -SUM can be
converted into an algorithm with an identical time improvement but low space
complexity as well. * -SUM is in deterministic time and space
, derandomizing an algorithm of Wang.
* A popular conjecture states that 3-SUM requires time on the
word-RAM. We show that the 3-SUM Conjecture is in fact equivalent to the
(seemingly weaker) conjecture that every -space algorithm for
-SUM requires at least time on the word-RAM.
* For , -SUM is in deterministic time and
space
Space--Time Tradeoffs for Subset Sum: An Improved Worst Case Algorithm
The technique of Schroeppel and Shamir (SICOMP, 1981) has long been the most
efficient way to trade space against time for the SUBSET SUM problem. In the
random-instance setting, however, improved tradeoffs exist. In particular, the
recently discovered dissection method of Dinur et al. (CRYPTO 2012) yields a
significantly improved space--time tradeoff curve for instances with strong
randomness properties. Our main result is that these strong randomness
assumptions can be removed, obtaining the same space--time tradeoffs in the
worst case. We also show that for small space usage the dissection algorithm
can be almost fully parallelized. Our strategy for dealing with arbitrary
instances is to instead inject the randomness into the dissection process
itself by working over a carefully selected but random composite modulus, and
to introduce explicit space--time controls into the algorithm by means of a
"bailout mechanism"
Generic Decoding of Restricted Errors
Several recently proposed code-based cryptosystems base their security on a
slightly generalized version of the classical (syndrome) decoding problem.
Namely, in the so-called restricted (syndrome) decoding problem, the error
values stem from a restricted set. In this paper, we propose new generic
decoders, that are inspired by subset sum solvers and tailored to the new
setting. The introduced algorithms take the restricted structure of the error
set into account in order to utilize the representation technique efficiently.
This leads to a considerable decrease in the security levels of recently
published code-based cryptosystems