We provide bounds on the efficiency of secure one-sided output two-party computation of arbitrary finite functions from trusted distributed randomness in the statistical case. From these results we derive bounds on the efficiency of protocols that use different variants of OT as a black-box. When applied to implementations of OT, these bounds generalize most known results to the statistical case. Our results hold in particular for transformations between a finite number of primitives and for any error. In the second part we study the efficiency of quantum protocols implementing OT. While most classical lower bounds for perfectly secure reductions of OT to distributed randomness still hold in the quantum setting, we present a statistically secure protocol that violates these bounds by an arbitrarily large factor. We then prove a weaker lower bound that does hold in the statistical quantum setting and implies that even quantum protocols cannot extend OT. Finally, we present two lower bounds for reductions of OT to commitments and a protocol based on string commitments that is optimal with respect to both of these bounds.We provide bounds on the efficiency of secure one-sided output two-party computation of arbitrary finite functions from trusted distributed randomness in the statistical case. From these results we derive bounds on the efficiency of protocols that use different variants of OT as a black-box. When applied to implementations of OT, these bounds generalize most known results to the statistical case. Our results hold in particular for transformations between a finite number of primitives and for any error. In the second part we study the efficiency of quantum protocols implementing OT. While most classical lower bounds for perfectly secure reductions of OT to distributed randomness still hold in the quantum setting, we present a statistically secure protocol that violates these bounds by an arbitrarily large factor. We then prove a weaker lower bound that does hold in the statistical quantum setting and implies that even quantum protocols cannot extend OT. Finally, we present two lower bounds for reductions of OT to commitments and a protocol based on string commitments that is optimal with respect to both of these bounds.

Aug 17, 20166 citations

Lower Bounds for Quantum Secure Function Evaluation Reductions

Esther Hänggi, Severin Winkler

One-sided output secure function evaluation is a cryptographic primitive where the two mutually distrustful players, Alice and Bob, both have a private input to a bivariate function. Bob obtains the value of the function for the given inputs, while Alice receives no output. It is known that this primitive cannot be securely implemented if the two players only have access to noiseless classical and quantum communication. In this work, we first show that Bob can extract the function values for all his possible inputs from any implementation of a non-trivial function that is correct and preserves the privacy of Bob's input. Our result holds in the non-asymptotic setting where the players have finite resources and the error is a constant. Then we consider protocols for secure function evaluation in a setup where the two players have access to trusted distributed randomness as a resource. Building upon the first result, we prove a bound on the efficiency of such cryptographic reductions for any non-trivial function in terms of the conditional entropies of the trusted randomness. From this result, we can derive lower bounds on the number of instances of different variants of OT needed to securely implement a given function.One-sided output secure function evaluation is a cryptographic primitive where the two mutually distrustful players, Alice and Bob, both have a private input to a bivariate function. Bob obtains the value of the function for the given inputs, while Alice receives no output. It is known that this primitive cannot be securely implemented if the two players only have access to noiseless classical and quantum communication. In this work, we first show that Bob can extract the function values for all his possible inputs from any implementation of a non-trivial function that is correct and preserves the privacy of Bob's input. Our result holds in the non-asymptotic setting where the players have finite resources and the error is a constant. Then we consider protocols for secure function evaluation in a setup where the two players have access to trusted distributed randomness as a resource. Building upon the first result, we prove a bound on the efficiency of such cryptographic reductions for any non-trivial function in terms of the conditional entropies of the trusted randomness. From this result, we can derive lower bounds on the number of instances of different variants of OT needed to securely implement a given function.

arXiv:2405.12121May 20, 20240 citations

Randomized Oblivious Transfer for Secure Multiparty Computation in the Quantum Setting

Bruno Costa, Pedro Costa Branco, Manuel Goulão +2 more

Secure computation is a powerful cryptographic tool that encompasses the evaluation of any multivariate function with arbitrary inputs from mutually distrusting parties. The oblivious transfer primitive serves is a basic building block for the general task of secure multi-party computation. Therefore, analyzing the security in the universal composability framework becomes mandatory when dealing with multi-party computation protocols composed of oblivious transfer subroutines. Furthermore, since the required number of oblivious transfer instances scales with the size of the circuits, oblivious transfer remains as a bottleneck for large-scale multi-party computation implementations. Techniques that allow one to extend a small number of oblivious transfers into a larger one in an efficient way make use of the oblivious transfer variant called randomized oblivious transfer. In this work, we present randomized versions of two known oblivious transfer protocols, one quantum and another post-quantum with ring learning with an error assumption. We then prove their security in the quantum universal composability framework, in a common reference string model.Secure computation is a powerful cryptographic tool that encompasses the evaluation of any multivariate function with arbitrary inputs from mutually distrusting parties. The oblivious transfer primitive serves is a basic building block for the general task of secure multi-party computation. Therefore, analyzing the security in the universal composability framework becomes mandatory when dealing with multi-party computation protocols composed of oblivious transfer subroutines. Furthermore, since the required number of oblivious transfer instances scales with the size of the circuits, oblivious transfer remains as a bottleneck for large-scale multi-party computation implementations. Techniques that allow one to extend a small number of oblivious transfers into a larger one in an efficient way make use of the oblivious transfer variant called randomized oblivious transfer. In this work, we present randomized versions of two known oblivious transfer protocols, one quantum and another post-quantum with ring learning with an error assumption. We then prove their security in the quantum universal composability framework, in a common reference string model.

EntropyJul 31, 202112 citations

Expressiveness of definitions and efficiency of constructions in computational cryptography

Virgil D. Gligor, Jonathan I. Katz, Omer Horvitz

The computational treatment of cryptography, and indeed any scientific treatment of a problem, is marked by its definitional side and by it constructive side. Results in this thesis better our understanding of both: on one side, they characterize the extent to which computational definitions capture the security of the basic task of symmetric encryption; on the other, they provide explicit bounds on the efficiency of commitment and secure two-party computation constructions. Specifically: (1) We relate the formal and computational treatments of symmetric encryption, obtaining a precise characterization of computational schemes whose computational semantics imply their formal semantics. We prove that this characterization is strictly weaker than previously-identified notions, and show how it may be realized in a simpler, more efficient manner. (2) We provide lower-bounds on the number of times a one-way permutation needs to be invoked (as a black-box) in order to construct statistically-binding commitments. Our bounds are tight for the case of perfectly-binding schemes. (3) We show that the secure computation of any two-party functionality can be performed in an optimal two rounds of communication even in a setting that accounts for concurrent execution with other protocols (i.e., the Universal Composability framework). Here, we rely on the assumption that parties have access to a common reference string; some sort of setup is known to be necessary.The computational treatment of cryptography, and indeed any scientific treatment of a problem, is marked by its definitional side and by it constructive side. Results in this thesis better our understanding of both: on one side, they characterize the extent to which computational definitions capture the security of the basic task of symmetric encryption; on the other, they provide explicit bounds on the efficiency of commitment and secure two-party computation constructions. Specifically: (1) We relate the formal and computational treatments of symmetric encryption, obtaining a precise characterization of computational schemes whose computational semantics imply their formal semantics. We prove that this characterization is strictly weaker than previously-identified notions, and show how it may be realized in a simpler, more efficient manner. (2) We provide lower-bounds on the number of times a one-way permutation needs to be invoked (as a black-box) in order to construct statistically-binding commitments. Our bounds are tight for the case of perfectly-binding schemes. (3) We show that the secure computation of any two-party functionality can be performed in an optimal two rounds of communication even in a setting that accounts for concurrent execution with other protocols (i.e., the Universal Composability framework). Here, we rely on the assumption that parties have access to a common reference string; some sort of setup is known to be necessary.

Jan 1, 20071 citation

A Constant Lower Bound for Any Quantum Protocol for Secure Function Evaluation

Osborn, Sarah A., Jamie Sikora

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.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.

arXiv:2203.08268Leibniz international proceedings in informaticsJan 1, 20221 citation

Secure Quantum Two-Party Computation: Impossibility and Constructions.

Michele Ciampi, Alexandru Cojocaru, Elham Kashefi +1 more

Secure two-party computation considers the problem of two parties computing a joint function of their private inputs without revealing anything beyond the output of the computation. In this work, we take the first steps towards understanding the setting in which the two parties want to evaluate a joint quantum functionality while using only a classical channel between them. Our first result indicates that it is in general impossible to realize a two-party quantum functionality against malicious adversaries with black-box simulation, relying only on classical channels. The negative result stems from reducing the existence of a black-box simulator to an extractor for classical proof of quantum knowledge, which in turn leads to violation of the quantum no-cloning. Next, we introduce the notion of oblivious quantum function evaluation (OQFE). An OQFE is a two-party quantum cryptographic primitive with one fully classical party (Alice) whose input is (a classical description of a) quantum unitary, $U$ , and a quantum party (Bob) whose input is a quantum state, $\psi$ . In particular, Alice receives a classical output corresponding to the measurement of $U(\psi)$ while Bob receives no output. In OQFE, Bob remains oblivious to Alice's input, while Alice learns nothing about $\psi$ more than what can be learned from the output. We present two constructions, one secure against semi-honest parties and the other against malicious parties. Due to the no-go result mentioned above, we consider what is arguably the best possible notion obtainable in our model concerning malicious adversaries: one-sided simulation security. Our protocol relies on the assumption of injective homomorphic trapdoor OWFs, which in turn rely on the LWE problem. As a result, we put forward a first, simple and modular, construction of one-sided quantum two-party computation and quantum oblivious transfer over classical networks.Secure two-party computation considers the problem of two parties computing a joint function of their private inputs without revealing anything beyond the output of the computation. In this work, we take the first steps towards understanding the setting in which the two parties want to evaluate a joint quantum functionality while using only a classical channel between them. Our first result indicates that it is in general impossible to realize a two-party quantum functionality against malicious adversaries with black-box simulation, relying only on classical channels. The negative result stems from reducing the existence of a black-box simulator to an extractor for classical proof of quantum knowledge, which in turn leads to violation of the quantum no-cloning. Next, we introduce the notion of oblivious quantum function evaluation (OQFE). An OQFE is a two-party quantum cryptographic primitive with one fully classical party (Alice) whose input is (a classical description of a) quantum unitary, $U$, and a quantum party (Bob) whose input is a quantum state, $\psi$. In particular, Alice receives a classical output corresponding to the measurement of $U(\psi)$ while Bob receives no output. In OQFE, Bob remains oblivious to Alice's input, while Alice learns nothing about $\psi$ more than what can be learned from the output. We present two constructions, one secure against semi-honest parties and the other against malicious parties. Due to the no-go result mentioned above, we consider what is arguably the best possible notion obtainable in our model concerning malicious adversaries: one-sided simulation security. Our protocol relies on the assumption of injective homomorphic trapdoor OWFs, which in turn rely on the LWE problem. As a result, we put forward a first, simple and modular, construction of one-sided quantum two-party computation and quantum oblivious transfer over classical networks.

Jan 1, 20201 citation

U

, and a quantum party (Bob) whose input is a quantum state,

\psi

. In particular, Alice receives a classical output corresponding to the measurement of

U(\psi)

while Bob receives no output. In OQFE, Bob remains oblivious to Alice's input, while Alice learns nothing about

\psi

more than what can be learned from the output. We present two constructions, one secure against semi-honest parties and the other against malicious parties. Due to the no-go result mentioned above, we consider what is arguably the best possible notion obtainable in our model concerning malicious adversaries: one-sided simulation security. Our protocol relies on the assumption of injective homomorphic trapdoor OWFs, which in turn rely on the LWE problem. As a result, we put forward a first, simple and modular, construction of one-sided quantum two-party computation and quantum oblivious transfer over classical networks.