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Oblivious Polynomial Evaluation and Secure Set-Intersection from Algebraic PRFs.

Carmit Hazay

Published Jan 1, 2015• 3 citations
OpenAlex

Abstract

In this paper we study the two fundamental functionalities oblivious polynomial evaluation in the exponent and set-intersection, and introduce a new technique for designing efficient secure protocols for these problems (and others). Our starting point is the [BGV11] technique (CRYPTO 2011) for verifiable delegation of polynomial evaluations, using algebraic PRFs. We use this tool, that is useful to achieve verifiability in the outsourced setting, in order to achieve privacy in the standard two-party setting. Our results imply new simple and efficient oblivious polynomial evaluation (OPE) protocols. We further show that our OPE protocols are readily used for secure set-intersection, implying much simpler protocols in the plain model. As a side result, we demonstrate the usefulness of algebraic PRFs for various search functionalities, such as keyword search and oblivious transfer with adaptive queries. Our protocols are secure under full simulation-based definitions in the presence of malicious adversaries.

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Efficient Oblivious Polynomial Evaluation with Simulation-Based Security.

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The study of secure multiparty computation has yielded powerful feasibility results showing that any efficient functionality can be securely computed in the presence of malicious adversaries. Despite this, there are few problems of specific interest for which we have highly efficient protocols that are secure in the presence of malicious adversaries under full simulation based definitions (following the ideal/real model paradigm). Due to the difficulties of constructing such protocols, many researchers have resorted to weaker definitions of security and weaker adversary models. In this paper, we construct highly efficient protocols for the well-studied problem of oblivious polynomial evaluation. Our protocol is secure under standard cryptographic assumptions for the settings of malicious adversaries, and readily transform to protocols that are secure under universal composability and in the presence of covert adversaries. Our protocol is constant round and requires O(d · s) exponentiations, where d is the degree of the polynomial and s is a statistical security parameter (that should equal about 160 in practice).

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Efficient Protocols for Set Intersection and Pattern Matching with Security Against Malicious and Covert Adversaries.

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Abstract. In this paper we construct efficient secure protocols for set intersection and pattern matching. Our protocols for securely computing the set intersection functionality are based on secure pseudorandom function evaluations, in contrast to previous protocols that used secure polynomial evaluation. In addition to the above, we also use secure pseudorandom function evaluation in order to achieve secure pattern matching. In this case, we utilize specific properties of the Naor-Reingold pseudorandom function in order to achieve high efficiency. Our results are presented in two adversary models. Our protocol for secure pattern matching and one of our protocols for set intersection achieve security against malicious adversaries under a relaxed definition where one corruption case is simulatable and for the other only privacy (formalized through indistinguishability) is guaranteed. We also present a protocol for set intersection that is fully simulatable in the model of covert adversaries. Loosely speaking, this means that a malicious adversary can cheat, but will then be caught with good probability. 1

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Oblivious Polynomial Evaluation

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The problem of two-party oblivious polynomial evaluation(OPE)is studied,where one party(Alice)has a polynomial P(x)and the other party(Bob)with an input x wants to learn P(x)in such an oblivious way that Bob obtains P(x)without learning any additional information about P except what is implied by P(x)and Alice does not know Bob's input x.The former OPE protocols are based on an intractability assumption except for OT protocols.In fact,evaluating P(x)is equivalent to computing the product of the coefficient vectors(a0,...,an)and(1,...,xn).Using this idea,an efficient scale product protocol of two vectors is proposed first and then two OPE protocols are presented which do not need any other cryptographic assumption except for OT protocol.Compared with the existing OPE protocol,another characteristic of the proposed protocols is the degree of the polynomial is private.Another OPE protocol works in case of existence of untrusted third party.

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In this paper we focus on protocols for private set intersection (PSI), through which two parties, each holding a set of inputs drawn from a ground set, jointly compute the intersection of their sets. Ideally, no further information than which elements are actually shared is compromised to the other party, yet the input set sizes are often considered as admissible leakage.In the unconditional setting we evidence that PSI is impossible to realize and that unconditionally secure size-hiding PSI is possible assuming a set-up authority is present in an set up phase. In the computational setting we give a generic construction using smooth projective hash functions for languages derived from perfectly-binding commitments. Further, we give two size-hiding constructions: the first one is theoretical and evidences the equivalence between PSI, oblivious transfer and the secure computation of the AND function. The second one is a twist on the oblivious polynomial evaluation construction of Freedman et al. from EUROCRYPT 2004. We further sketch a generalization of the latter using algebraic-geometric techniques. Finally, assuming again there is a set-up authority (yet not necessarily trusted) we present very simple and efficient constructions that only hide the size of the client's set.

Advances in Mathematics of Communications•Jan 1, 2017•3 citations
In this paper we study the two fundamental functionalities oblivious polynomial evaluation in the exponent and set-intersection, and introduce a new technique for designing efficient secure protocols for these problems (and others). Our starting point is the [BGV11] technique (CRYPTO 2011) for verifiable delegation of polynomial evaluations, using algebraic PRFs. We use this tool, that is useful to achieve verifiability in the outsourced setting, in order to achieve privacy in the standard two-party setting. Our results imply new simple and efficient oblivious polynomial evaluation (OPE) protocols. We further show that our OPE protocols are readily used for secure set-intersection, implying much simpler protocols in the plain model. As a side result, we demonstrate the usefulness of algebraic PRFs for various search functionalities, such as keyword search and oblivious transfer with adaptive queries. Our protocols are secure under full simulation-based definitions in the presence of malicious adversaries.
The study of secure multiparty computation has yielded powerful feasibility results showing that any efficient functionality can be securely computed in the presence of malicious adversaries. Despite this, there are few problems of specific interest for which we have highly efficient protocols that are secure in the presence of malicious adversaries under full simulation based definitions (following the ideal/real model paradigm). Due to the difficulties of constructing such protocols, many researchers have resorted to weaker definitions of security and weaker adversary models. In this paper, we construct highly efficient protocols for the well-studied problem of oblivious polynomial evaluation. Our protocol is secure under standard cryptographic assumptions for the settings of malicious adversaries, and readily transform to protocols that are secure under universal composability and in the presence of covert adversaries. Our protocol is constant round and requires O(d · s) exponentiations, where d is the degree of the polynomial and s is a statistical security parameter (that should equal about 160 in practice).
Abstract. In this paper we construct efficient secure protocols for set intersection and pattern matching. Our protocols for securely computing the set intersection functionality are based on secure pseudorandom function evaluations, in contrast to previous protocols that used secure polynomial evaluation. In addition to the above, we also use secure pseudorandom function evaluation in order to achieve secure pattern matching. In this case, we utilize specific properties of the Naor-Reingold pseudorandom function in order to achieve high efficiency. Our results are presented in two adversary models. Our protocol for secure pattern matching and one of our protocols for set intersection achieve security against malicious adversaries under a relaxed definition where one corruption case is simulatable and for the other only privacy (formalized through indistinguishability) is guaranteed. We also present a protocol for set intersection that is fully simulatable in the model of covert adversaries. Loosely speaking, this means that a malicious adversary can cheat, but will then be caught with good probability. 1
The problem of two-party oblivious polynomial evaluation(OPE)is studied,where one party(Alice)has a polynomial P(x)and the other party(Bob)with an input x wants to learn P(x)in such an oblivious way that Bob obtains P(x)without learning any additional information about P except what is implied by P(x)and Alice does not know Bob's input x.The former OPE protocols are based on an intractability assumption except for OT protocols.In fact,evaluating P(x)is equivalent to computing the product of the coefficient vectors(a0,...,an)and(1,...,xn).Using this idea,an efficient scale product protocol of two vectors is proposed first and then two OPE protocols are presented which do not need any other cryptographic assumption except for OT protocol.Compared with the existing OPE protocol,another characteristic of the proposed protocols is the degree of the polynomial is private.Another OPE protocol works in case of existence of untrusted third party.
We revisit the problem of constructing efficient secure two-party protocols for the problems of setintersection and set-union, focusing on the model of malicious parties. Our main results are constantround protocols that exhibit linear communication and a (practically) linear number of exponentiations with simulation based security. In the heart of these constructions is a technique based on a combination of a perfectly hiding commitment and an oblivious pseudorandom function evaluation protocol. Our protocols readily transform into protocols that are UC-secure, and we discuss how to perform these transformations. Keywords: Secure two-party computation, Simulation-based security, Set-intersection, Set-union, Oblivious pseudorandom function evaluation.
We describe efficient constructions for two oblivious twoparty computation problems: l-out-of-N Oblivious Transfer &d 'Oblivious Poly&nial Evaluation.The oblivious polynomial evaluation protocol is based on a new intractability assumption which is closely related to noisy polynomial re construction.A direct corollary of the l-out-of-N OT protccol is an efficient transformation of any Private Information Retrieval (PIR) protocol to a Symmetric PIR (SPIR) prc-tow1 without increasing the number of databases.The new construction for l-out-of-N OT is highly efficient -it requires only log N executions of a l-out-of-2 OT protocol.We also present a construction for k-out-of-N OT which is more efficient than k repetitions of l-out-of-N OT.The efficiency of the new OT protocols makes them useful for a variety of applications.These include oblivious sampling which can be used to securely compare the sizes of web search engines, protocols for privately solving the list intersection problem and for mutually authenticated key exchange based on (possibly weak) passwords, and protocols for anonymity preserving web usage metering.
In this paper we focus on protocols for private set intersection (PSI), through which two parties, each holding a set of inputs drawn from a ground set, jointly compute the intersection of their sets. Ideally, no further information than which elements are actually shared is compromised to the other party, yet the input set sizes are often considered as admissible leakage.In the unconditional setting we evidence that PSI is impossible to realize and that unconditionally secure size-hiding PSI is possible assuming a set-up authority is present in an set up phase. In the computational setting we give a generic construction using smooth projective hash functions for languages derived from perfectly-binding commitments. Further, we give two size-hiding constructions: the first one is theoretical and evidences the equivalence between PSI, oblivious transfer and the secure computation of the AND function. The second one is a twist on the oblivious polynomial evaluation construction of Freedman et al. from EUROCRYPT 2004. We further sketch a generalization of the latter using algebraic-geometric techniques. Finally, assuming again there is a set-up authority (yet not necessarily trusted) we present very simple and efficient constructions that only hide the size of the client's set.