445 research outputs found
A Quantum-Proof Non-Malleable Extractor, With Application to Privacy Amplification against Active Quantum Adversaries
In privacy amplification, two mutually trusted parties aim to amplify the
secrecy of an initial shared secret in order to establish a shared private
key by exchanging messages over an insecure communication channel. If the
channel is authenticated the task can be solved in a single round of
communication using a strong randomness extractor; choosing a quantum-proof
extractor allows one to establish security against quantum adversaries.
In the case that the channel is not authenticated, Dodis and Wichs (STOC'09)
showed that the problem can be solved in two rounds of communication using a
non-malleable extractor, a stronger pseudo-random construction than a strong
extractor.
We give the first construction of a non-malleable extractor that is secure
against quantum adversaries. The extractor is based on a construction by Li
(FOCS'12), and is able to extract from source of min-entropy rates larger than
. Combining this construction with a quantum-proof variant of the
reduction of Dodis and Wichs, shown by Cohen and Vidick (unpublished), we
obtain the first privacy amplification protocol secure against active quantum
adversaries
Optimal Multi-Pass Lower Bounds for MST in Dynamic Streams
The seminal work of Ahn, Guha, and McGregor in 2012 introduced the graph
sketching technique and used it to present the first streaming algorithms for
various graph problems over dynamic streams with both insertions and deletions
of edges. This includes algorithms for cut sparsification, spanners, matchings,
and minimum spanning trees (MSTs). These results have since been improved or
generalized in various directions, leading to a vastly rich host of efficient
algorithms for processing dynamic graph streams.
A curious omission from the list of improvements has been the MST problem.
The best algorithm for this problem remains the original AGM algorithm that for
every integer , uses space in passes on -vertex
graphs, and thus achieves the desired semi-streaming space of at
a relatively high cost of passes. On the other
hand, no lower bounds beyond a folklore one-pass lower bound is known for this
problem.
We provide a simple explanation for this lack of improvements: The AGM
algorithm for MSTs is optimal for the entire range of its number of passes! We
prove that even for the simplest decision version of the problem -- deciding
whether the weight of MSTs is at least a given threshold or not -- any -pass
dynamic streaming algorithm requires space. This implies
that semi-streaming algorithms do need
passes.
Our result relies on proving new multi-round communication complexity lower
bounds for a variant of the universal relation problem that has been
instrumental in proving prior lower bounds for single-pass dynamic streaming
algorithms. The proof also involves proving new composition theorems in
communication complexity, including majority lemmas and multi-party XOR lemmas,
via information complexity approaches
A Computational Tsirelson's Theorem for the Value of Compiled XOR Games
Nonlocal games are a foundational tool for understanding entanglement and
constructing quantum protocols in settings with multiple spatially separated
quantum devices. In this work, we continue the study initiated by Kalai et al.
(STOC '23) of compiled nonlocal games, played between a classical verifier and
a single cryptographically limited quantum device. Our main result is that the
compiler proposed by Kalai et al. is sound for any two-player XOR game. A
celebrated theorem of Tsirelson shows that for XOR games, the quantum value is
exactly given by a semidefinite program, and we obtain our result by showing
that the SDP upper bound holds for the compiled game up to a negligible error
arising from the compilation. This answers a question raised by Natarajan and
Zhang (FOCS '23), who showed soundness for the specific case of the CHSH game.
Using our techniques, we obtain several additional results, including (1) tight
bounds on the compiled value of parallel-repeated XOR games, (2) operator
self-testing statements for any compiled XOR game, and (3) a ``nice"
sum-of-squares certificate for any XOR game, from which operator rigidity is
manifest
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