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**Published in:** LIPIcs, Volume 232, 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)

Secure function evaluation is a two-party cryptographic primitive where Bob computes a function of Alice’s and his respective inputs, and both hope to keep their inputs private from the other party. It has been proven that perfect (or near perfect) security is impossible, even for quantum protocols. We generalize this no-go result by exhibiting a constant lower bound on the cheating probabilities for any quantum protocol for secure function evaluation, and present many applications from oblivious transfer to the millionaire’s problem. Constant lower bounds are of practical interest since they imply the impossibility to arbitrarily amplify the security of quantum protocols by any means.

Sarah A. Osborn and Jamie Sikora. A Constant Lower Bound for Any Quantum Protocol for Secure Function Evaluation. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 232, pp. 8:1-8:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{osborn_et_al:LIPIcs.TQC.2022.8, author = {Osborn, Sarah A. and Sikora, Jamie}, title = {{A Constant Lower Bound for Any Quantum Protocol for Secure Function Evaluation}}, booktitle = {17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022)}, pages = {8:1--8:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-237-2}, ISSN = {1868-8969}, year = {2022}, volume = {232}, editor = {Le Gall, Fran\c{c}ois and Morimae, Tomoyuki}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2022.8}, URN = {urn:nbn:de:0030-drops-165151}, doi = {10.4230/LIPIcs.TQC.2022.8}, annote = {Keywords: Quantum cryptography, security analysis, secure function evaluation} }

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**Published in:** LIPIcs, Volume 158, 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)

Oblivious transfer is a cryptographic primitive where Alice has two bits and Bob wishes to learn some function of them. Ideally, Alice should not learn Bob’s desired function choice and Bob should not learn any more than logically implied by the function value. While decent quantum protocols for this task are known, many quickly become insecure if an adversary were to control the quantum devices used in the implementation of the protocol. Here we present how some existing protocols fail in this device-independent framework, and give a fully-device independent quantum protocol for XOR oblivious transfer which is provably more secure than any classical protocol.

Srijita Kundu, Jamie Sikora, and Ernest Y.-Z. Tan. A Device-Independent Protocol for XOR Oblivious Transfer. In 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 158, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kundu_et_al:LIPIcs.TQC.2020.12, author = {Kundu, Srijita and Sikora, Jamie and Tan, Ernest Y.-Z.}, title = {{A Device-Independent Protocol for XOR Oblivious Transfer}}, booktitle = {15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)}, pages = {12:1--12:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-146-7}, ISSN = {1868-8969}, year = {2020}, volume = {158}, editor = {Flammia, Steven T.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2020.12}, URN = {urn:nbn:de:0030-drops-127579}, doi = {10.4230/LIPIcs.TQC.2020.12}, annote = {Keywords: Quantum cryptography, device independence, oblivious transfer, semidefinite programming, security analysis} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

The polynomial-time hierarchy (PH) has proven to be a powerful tool for providing separations in computational complexity theory (modulo standard conjectures such as PH does not collapse). Here, we study whether two quantum generalizations of PH can similarly prove separations in the quantum setting. The first generalization, QCPH, uses classical proofs, and the second, QPH, uses quantum proofs. For the former, we show quantum variants of the Karp-Lipton theorem and Toda's theorem. For the latter, we place its third level, Q Sigma_3, into NEXP using the Ellipsoid Method for efficiently solving semidefinite programs. These results yield two implications for QMA(2), the variant of Quantum Merlin-Arthur (QMA) with two unentangled proofs, a complexity class whose characterization has proven difficult. First, if QCPH=QPH (i.e., alternating quantifiers are sufficiently powerful so as to make classical and quantum proofs "equivalent"), then QMA(2) is in the Counting Hierarchy (specifically, in P^{PP^{PP}}). Second, unless QMA(2)= Q Sigma_3 (i.e., alternating quantifiers do not help in the presence of "unentanglement"), QMA(2) is strictly contained in NEXP.

Sevag Gharibian, Miklos Santha, Jamie Sikora, Aarthi Sundaram, and Justin Yirka. Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2). In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 58:1-58:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gharibian_et_al:LIPIcs.MFCS.2018.58, author = {Gharibian, Sevag and Santha, Miklos and Sikora, Jamie and Sundaram, Aarthi and Yirka, Justin}, title = {{Quantum Generalizations of the Polynomial Hierarchy with Applications to QMA(2)}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {58:1--58:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.58}, URN = {urn:nbn:de:0030-drops-96409}, doi = {10.4230/LIPIcs.MFCS.2018.58}, annote = {Keywords: Complexity Theory, Quantum Computing, Polynomial Hierarchy, Semidefinite Programming, QMA(2), Quantum Complexity} }

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**Published in:** LIPIcs, Volume 73, 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)

We introduce a definition of the fidelity function for multi-round quantum strategies, which we call the strategy fidelity, that is a generalization of the fidelity function for quantum states. We provide many interesting properties of the strategy fidelity including a Fuchs-van de Graaf relationship with the strategy norm. We illustrate an operational interpretation of the strategy fidelity in the spirit of Uhlmann's Theorem and discuss its application to the security analysis of quantum protocols for interactive cryptographic tasks such as bit-commitment and oblivious string transfer. Our analysis is very general in the sense that the actions of the protocol need not be fully specified, which is in stark contrast to most other security proofs. Lastly, we provide a semidefinite programming formulation of the strategy fidelity.

Gus Gutoski, Ansis Rosmanis, and Jamie Sikora. Fidelity of Quantum Strategies with Applications to Cryptography. In 12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 73, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gutoski_et_al:LIPIcs.TQC.2017.8, author = {Gutoski, Gus and Rosmanis, Ansis and Sikora, Jamie}, title = {{Fidelity of Quantum Strategies with Applications to Cryptography}}, booktitle = {12th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2017)}, pages = {8:1--8:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-034-7}, ISSN = {1868-8969}, year = {2018}, volume = {73}, editor = {Wilde, Mark M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2017.8}, URN = {urn:nbn:de:0030-drops-85830}, doi = {10.4230/LIPIcs.TQC.2017.8}, annote = {Keywords: Quantum strategies, cryptography, fidelity, semidefinite programming} }

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**Published in:** LIPIcs, Volume 61, 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)

Die-rolling is the cryptographic task where two mistrustful, remote parties wish to generate a random D-sided die-roll over a communication channel. Optimal quantum protocols for this task have been given by Aharon and Silman (New Journal of Physics, 2010) but are based on optimal weak coin-flipping protocols which are currently very complicated and not very well understood. In this paper, we first present very simple classical protocols for die-rolling which have decent (and sometimes optimal) security which is in stark contrast to coin-flipping, bit-commitment, oblivious transfer, and many other two-party cryptographic primitives. We also present quantum protocols based on the idea of integer-commitment, a generalization of bit-commitment, where one wishes to commit to an integer. We analyze these protocols using semidefinite programming and finally give protocols which are very close to Kitaev's lower bound for any D >= 3.

Jamie Sikora. Simple, Near-Optimal Quantum Protocols for Die-Rolling. In 11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 61, pp. 4:1-4:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{sikora:LIPIcs.TQC.2016.4, author = {Sikora, Jamie}, title = {{Simple, Near-Optimal Quantum Protocols for Die-Rolling}}, booktitle = {11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)}, pages = {4:1--4:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-019-4}, ISSN = {1868-8969}, year = {2016}, volume = {61}, editor = {Broadbent, Anne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2016.4}, URN = {urn:nbn:de:0030-drops-66851}, doi = {10.4230/LIPIcs.TQC.2016.4}, annote = {Keywords: Quantum Cryptography, Semidefinite Programming, Die-Rolling, Integer-Commitment} }

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**Published in:** LIPIcs, Volume 27, 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)

Random Access Codes is an information task that has been extensively studied and found many applications in quantum information. In this scenario, Alice receives an n-bit string x, and wishes to encode x into a quantum state rho_x, such that Bob, when receiving the state rho_x, can choose any bit i in [n] and recover the input bit x_i with high probability. Here we study a variant called parity-oblivious random acres codes, where we impose the cryptographic property that Bob cannot infer any information about the parity of any subset of bits of the input, apart form the single bits x_i.
We provide the optimal quantum parity-oblivious random access codes and show that they are asymptotically better than the optimal classical ones. For this, we relate such encodings to a non-local game and provide tight bounds for the success probability of the non-local game via semi-definite programming. Our results provide a large non-contextuality inequality violation and resolve the main open question in [Spekkens et al., Phys. Review Letters, 2009].

André Chailloux, Iordanis Kerenidis, Srijita Kundu, and Jamie Sikora. Optimal Bounds for Parity-Oblivious Random Access Codes with Applications. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 27, pp. 76-87, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{chailloux_et_al:LIPIcs.TQC.2014.76, author = {Chailloux, Andr\'{e} and Kerenidis, Iordanis and Kundu, Srijita and Sikora, Jamie}, title = {{Optimal Bounds for Parity-Oblivious Random Access Codes with Applications}}, booktitle = {9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014)}, pages = {76--87}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-73-6}, ISSN = {1868-8969}, year = {2014}, volume = {27}, editor = {Flammia, Steven T. and Harrow, Aram W.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2014.76}, URN = {urn:nbn:de:0030-drops-48084}, doi = {10.4230/LIPIcs.TQC.2014.76}, annote = {Keywords: quantum information theory, contextuality, semidefinite programming} }

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**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

Oblivious transfer is a fundamental primitive in cryptography. While perfect information theoretic security is impossible, quantum oblivious transfer protocols can limit the dishonest players' cheating. Finding the optimal security parameters in such protocols is an important open question. In this paper we show that every 1-out-of-2 oblivious transfer protocol allows a dishonest party to cheat with probability bounded below by a constant strictly larger than $1/2$. Alice's cheating is defined as her probability of guessing Bob's index, and Bob's cheating is defined as his probability of guessing both input bits of Alice. In our proof, we relate these cheating probabilities to the cheating probabilities of a coin flipping protocol and conclude by using Kitaev's coin flipping lower bound. Then, we present an oblivious transfer protocol with two messages and cheating probabilities at most $3/4$. Last, we extend Kitaev's semidefinite programming formulation to more general primitives, where the security is against a dishonest player trying to force the outcome of the other player, and prove optimal lower
and upper bounds for them.

André Chailloux, Iordanis Kerenidis, and Jamie Sikora. Lower bounds for Quantum Oblivious Transfer. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 157-168, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{chailloux_et_al:LIPIcs.FSTTCS.2010.157, author = {Chailloux, Andr\'{e} and Kerenidis, Iordanis and Sikora, Jamie}, title = {{Lower bounds for Quantum Oblivious Transfer}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {157--168}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.157}, URN = {urn:nbn:de:0030-drops-28613}, doi = {10.4230/LIPIcs.FSTTCS.2010.157}, annote = {Keywords: quantum oblivious transfer, coin flipping protocol, semidefinite programming} }