48,896 research outputs found
On Finding Quantum Multi-collisions
A -collision for a compressing hash function is a set of distinct
inputs that all map to the same output. In this work, we show that for any
constant , quantum
queries are both necessary and sufficient to achieve a -collision with
constant probability. This improves on both the best prior upper bound
(Hosoyamada et al., ASIACRYPT 2017) and provides the first non-trivial lower
bound, completely resolving the problem
Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision
The quantum query complexity of Boolean matrix multiplication is typically
studied as a function of the matrix dimension, n, as well as the number of 1s
in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values
of \ell. This is an improvement over previous algorithms for all values of
\ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps
n^2, there is an \Omega(n\sqrt{\ell}) lower bound for this problem, showing
that our algorithm is essentially tight.
We first reduce Boolean matrix multiplication to several instances of graph
collision. We then provide an algorithm that takes advantage of the fact that
the underlying graph in all of our instances is very dense to find all graph
collisions efficiently
Quantum Query Complexity of Multilinear Identity Testing
Motivated by the quantum algorithm in \cite{MN05} for testing commutativity
of black-box groups, we study the following problem: Given a black-box finite
ring where is an additive
generating set for and a multilinear polynomial over
also accessed as a black-box function (where we allow the
indeterminates to be commuting or noncommuting), we study the
problem of testing if is an \emph{identity} for the ring . More
precisely, the problem is to test if for all .
We give a quantum algorithm with query complexity assuming . Towards a lower bound,
we also discuss a reduction from a version of -collision to this problem.
We also observe a randomized test with query complexity and constant
success probability and a deterministic test with query complexity.Comment: 12 page
On Quantum Query Complexities of Collision-Finding in Non-Uniform Random Functions
Collision resistance and collision finding are now extensively exploited in Cryptography, especially in the case of quantum computing. For any function with uniformly distributed over , Zhandry has shown that the number of queries is both necessary and sufficient for finding a collision in with constant probability. However, there is still a gap between the upper and the lower bounds of query complexity in general non-uniform distributions.
In this paper, we investigate the quantum query complexity of collision-finding problem with respect to general non-uniform distributions. Inspired by previous work, we pose the concept of collision domain and a new parameter that heavily depends on the underlying non-uniform distribution. We then present a quantum algorithm that uses quantum queries to find a collision for any non-uniform random function. By making a transformation of a problem in non-uniform setting into a problem in uniform setting, we are also able to show that quantum queries are necessary in collision-finding in any non-uniform random function.
The upper bound and the lower bound in this work indicates that the proposed algorithm is nearly optimal with query complexity in general non-uniform case
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