34,756 research outputs found
Quantum Algorithm for the Collision Problem
In this note, we give a quantum algorithm that finds collisions in arbitrary
r-to-one functions after only O((N/r)^(1/3)) expected evaluations of the
function. Assuming the function is given by a black box, this is more efficient
than the best possible classical algorithm, even allowing probabilism. We also
give a similar algorithm for finding claws in pairs of functions. Furthermore,
we exhibit a space-time tradeoff for our technique. Our approach uses Grover's
quantum searching algorithm in a novel way.Comment: 8 pages, LaTeX2
The NISQ Complexity of Collision Finding
Collision-resistant hashing, a fundamental primitive in modern cryptography,
ensures that there is no efficient way to find distinct inputs that produce the
same hash value. This property underpins the security of various cryptographic
applications, making it crucial to understand its complexity. The complexity of
this problem is well-understood in the classical setting and
queries are needed to find a collision. However, the advent of quantum
computing has introduced new challenges since quantum adversaries
\unicode{x2013} equipped with the power of quantum queries \unicode{x2013}
can find collisions much more efficiently. Brassard, H\"oyer and Tapp and
Aaronson and Shi established that full-scale quantum adversaries require
queries to find a collision, prompting a need for longer hash
outputs, which impacts efficiency in terms of the key lengths needed for
security.
This paper explores the implications of quantum attacks in the
Noisy-Intermediate Scale Quantum (NISQ) era. In this work, we investigate three
different models for NISQ algorithms and achieve tight bounds for all of them:
(1) A hybrid algorithm making adaptive quantum or classical queries but with
a limited quantum query budget, or
(2) A quantum algorithm with access to a noisy oracle, subject to a dephasing
or depolarizing channel, or
(3) A hybrid algorithm with an upper bound on its maximum quantum depth;
i.e., a classical algorithm aided by low-depth quantum circuits.
In fact, our results handle all regimes between NISQ and full-scale quantum
computers. Previously, only results for the pre-image search problem were known
for these models by Sun and Zheng, Rosmanis, Chen, Cotler, Huang and Li while
nothing was known about the collision finding problem.Comment: 40 pages; v2: title changed, major extension to other complexity
model
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
Low-gate Quantum Golden Collision Finding
International audienceThe golden collision problem asks us to find a single, special collision among the outputs of a pseudorandom function. This generalizes meet-in-the-middle problems, and is thus applicable in many contexts, such as cryptanalysis of the NIST post-quantum candidate SIKE. The main quantum algorithms for this problem are memory-intensive, and the costs of quantum memory may be very high. The quantum circuit model implies a linear cost for random access, which annihilates the exponential advantage of the previous quantum collision-finding algorithms over Grover's algorithm or classical van Oorschot-Wiener. Assuming that quantum memory is costly to access but free to maintain, we provide new quantum algorithms for the golden collision problem with high memory requirements but low gate costs. Under the assumption of a two-dimensional connectivity layout, we provide better quantum parallelization methods for generic and golden collision finding. This lowers the quantum security of the golden collision and meet-in-the-middle problems, including SIKE
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
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