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**Published in:** LIPIcs, Volume 199, 2nd Conference on Information-Theoretic Cryptography (ITC 2021)

The private simultaneous messages (PSM) model is a non-interactive version of the multiparty secure computation (MPC), which has been intensively studied to examine the communication cost of the secure computation. We consider its quantum counterpart, the private simultaneous quantum messages (PSQM) model, and examine the advantages of quantum communication and prior entanglement of this model.
In the PSQM model, k parties P₁,…,P_k initially share a common random string (or entangled states in a stronger setting), and they have private classical inputs x₁,…, x_k. Every P_i generates a quantum message from the private input x_i and the shared random string (entangled states), and then sends it to the referee R. Receiving the messages from the k parties, R computes F(x₁,…,x_k) from the messages. Then, R learns nothing except for F(x₁,…,x_k) as the privacy condition.
We obtain the following results for this PSQM model. (i) We demonstrate that the privacy condition inevitably increases the communication cost in the two-party PSQM model as well as in the classical case presented by Applebaum, Holenstein, Mishra, and Shayevitz [Journal of Cryptology(3), 916-953 (2020)]. In particular, we prove a lower bound (3-o(1))n of the communication complexity in PSQM protocols with a shared random string for random Boolean functions of 2n-bit input, which is larger than the trivial upper bound 2n of the communication complexity without the privacy condition. (ii) We demonstrate a factor two gap between the communication complexity of PSQM protocols with shared entangled states and with shared random strings by designing a multiparty PSQM protocol with shared entangled states for a total function that extends the two-party equality function. (iii) We demonstrate an exponential gap between the communication complexity of PSQM protocols with shared entangled states and with shared random strings for a two-party partial function.

Akinori Kawachi and Harumichi Nishimura. Communication Complexity of Private Simultaneous Quantum Messages Protocols. In 2nd Conference on Information-Theoretic Cryptography (ITC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 199, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kawachi_et_al:LIPIcs.ITC.2021.20, author = {Kawachi, Akinori and Nishimura, Harumichi}, title = {{Communication Complexity of Private Simultaneous Quantum Messages Protocols}}, booktitle = {2nd Conference on Information-Theoretic Cryptography (ITC 2021)}, pages = {20:1--20:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-197-9}, ISSN = {1868-8969}, year = {2021}, volume = {199}, editor = {Tessaro, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2021.20}, URN = {urn:nbn:de:0030-drops-143393}, doi = {10.4230/LIPIcs.ITC.2021.20}, annote = {Keywords: Communication complexity, private simultaneous messages, quantum protocols, secure multi-party computation} }

Document

**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

A Boolean Finite Synchronous Dynamical System (BFDS, for short) consists of a finite number of objects that each maintains a boolean state, where after individually receiving state assignments, the objects update their state with respect to object-specific time-independent boolean functions synchronously in discrete time steps.
The present paper studies the computational complexity of determining, given a boolean finite synchronous dynamical system,
a configuration, which is a boolean vector representing the states
of the objects, and a positive integer t, whether there exists another configuration from which the given configuration can be reached in t steps. It was previously shown that this problem, which we call the t-Predecessor Problem, is NP-complete even for t = 1
if the update function of an object is either the conjunction of
arbitrary fan-in or the disjunction of arbitrary fan-in.
This paper studies the computational complexity of the t-Predecessor Problem for a variety of sets of permissible update functions as well as for polynomially bounded t. It also studies the t-Garden-Of-Eden Problem, a variant of the t-Predecessor Problem that asks whether a configuration has a t-predecessor, which itself has no predecessor. The paper obtains complexity theoretical characterizations of all but one of these problems.

Akinori Kawachi, Mitsunori Ogihara, and Kei Uchizawa. Generalized Predecessor Existence Problems for Boolean Finite Dynamical Systems. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{kawachi_et_al:LIPIcs.MFCS.2017.8, author = {Kawachi, Akinori and Ogihara, Mitsunori and Uchizawa, Kei}, title = {{Generalized Predecessor Existence Problems for Boolean Finite Dynamical Systems}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {8:1--8:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.8}, URN = {urn:nbn:de:0030-drops-81078}, doi = {10.4230/LIPIcs.MFCS.2017.8}, annote = {Keywords: Computational complexity, dynamical systems, Garden of Eden, predecessor} }

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