Abstract We polish a recent cryptocomputing method of Ishai and Paskin from TCC 2007. More precisely, we show that every function can be cryptocomputed in communication, linear in the product of client’s input length and the length of the branching program, and computation, linear in the size of the branching program that computes it. The method is based on the existence of a communication-efficient (2, 1)-CPIR protocol. We give several nontrivial applications, including: (a) improvement on the communication of Lipmaa’s CPIR protocol, (b) a CPIR protocol with log-squared communication and sublinear server-computation by giving a secure function evaluation protocol for Boolean functions with similar performance, (c) a protocol for PIR-writing with low amortized complexity, (d) a selective private function evaluation (SPFE) protocol. We detail one application of SPFE that makes it possible to compute how similar is client’s input to an element in server’s database, without revealing any information to the server. For SPFE, we design a 4-message extension of the basic protocol that is efficient for a large class of functionalities.
Abstract We polish a recent cryptocomputing method of Ishai and Paskin from TCC 2007. More precisely, we show that every function can be cryptocomputed in communication, linear in the product of client’s input length and the length of the branching program, and computation, linear in the size of the branching program that computes it. The method is based on the existence of a communication-efficient (2, 1)-CPIR protocol. We give several nontrivial applications, including: (a) improvement on the communication of Lipmaa’s CPIR protocol, (b) a CPIR protocol with log-squared communication and sublinear server-computation by giving a secure function evaluation protocol for Boolean functions with similar performance, (c) a protocol for PIR-writing with low amortized complexity, (d) a selective private function evaluation (SPFE) protocol. We detail one application of SPFE that makes it possible to compute how similar is client’s input to an element in server’s database, without revealing any information to the server. For SPFE, we design a 4-message extension of the basic protocol that is efficient for a large class of functionalities.
In this dissertation, we address several issues that arise in protecting communication between parties, as well as in the area of secure function evaluation. Intuitively, the notion of secure function evaluation is clear and natural: several parties wish to compute some function of their inputs without revealing any information about the inputs, other than what is implied by the value of the function. Research included in this dissertation follows three main directions, briefly described below. The first direction (Chapters 3 and 4) is the design of efficient protocols for concrete functions of interest. Specifically, we present new, more efficient protocols for securely computing the Greater Than (GT) function on the inputs of two parties. Secure evaluation of GT is frequently needed in financial transactions. We introduce new primitives, which are convenient building blocks for more complex tasks, and generalize our GT solutions to satisfy them. Based on this, we construct secure auction protocols, protocols for determining whether an integer lies on an interval, and others. The second direction (Chapter 5) is a fundamental approach to secure evaluation of any function, given as a boolean circuit. We present a very efficient information-theoretic (IT) reduction from the problem of secure evaluation of a polysize formula (or, equivalently, a log-depth boolean circuit) to Oblivious Transfer (a fundamental well-researched cryptographic primitive). Our cost of evaluating each gate of the formula is quadratic in its depth, while in previous reductions it was exponential. Our constructions imply efficient one-round protocols for evaluation of polysize formulas on the players' inputs. We extend our solutions to evaluation of polysize circuits, at the cost of having only computational security. The third direction (Chapter 6) is research on key exchange (KE). In contrast with the previous two directions, here the goal is for two parties to protect their communication against eavesdropping and active interference of an external attacker. KE is a basic procedure, frequently used to establish secure channels between parties. It is a prerequisite to a large number of protocols, including those of the above two directions. We demonstrate a subtle flaw in a previous family of KE protocols and give new KE definitions for the following practical "bank" setting. Here, a server wishes to exchange a key with a client. They have a shared password, and the client carries a "bank card", capable of storing several cryptographic keys. Finally, we present new, more efficient KE protocols for this setting, and prove their security.
A secure function evaluation protocol allows two parties to jointly compute a function f(x; y) of their inputs in a manner not leaking more information than necessary. A major result in this field is: "any function f that can be computed using polynomial resources can be computed securely using polynomial resources" (where `resources' refers to communication and computation). This result follows by a general transformation from any circuit for f to a secure protocol that evaluates f . Although the resources used by protocols resulting from this transformation are polynomial in the circuit size, they are much higher (in general) than those required for an insecure computation of f . We propose a new methodology for designing secure protocols, utilizing the communication complexity tree (or branching program) representation of f . We start with an efficient (insecure) protocol for f and transform it into a secure protocol. In other words, "any function f that can be computed using communication complexity c can be can be computed securely using communication complexity that is polynomial in c and a security parameter". We show several simple applications of this new methodology resulting in protocols efficient either in communication or in computation. In particular, we exemplify a protocol for the "millionaires problem ", where two participants want to compare their values but reveal no other information. Our protocol is more efficient than previously known ones in either communication or computation. 1.
Development of computing devices with the proliferation of the Internet has prompted enormous opportunities for cooperative computation. These computations could occur between trusted or partially trusted partners, or even between competitors. Secure multi-party computation (MPC) protocols allow two or more parties to collaborate and compute a public functionality using their private inputs without the need for a trusted third-party. However, the generic solutions for MPC are not adequate for some particular cases where the function itself is also sensitive and required to be kept private. Private function evaluation (PFE) is a special case of MPC, where the function to be computed is known by only one party. PFE is useful in several real-life applications where an algorithm or a function itself needs to remain secret for reasons such as protecting intellectual property or security classification level. Recently, designing efficient PFE protocols have been a challenging and attractive task for cryptography researchers. iv In this dissertation, we mainly focus on improving two-party private function evaluation (2PFE) schemes. Our primary goal is enhancing the state-of-the-art by designing secure and cost-efficient 2PFE protocols for both symmetric and asymmetric cryptography based solutions. In this respect, we first aim to improve 2PFE protocols based on (mostly) symmetric cryptographic primitives. We look back at the seminal PFE framework presented by Mohassel and Sadeghian at Eurocrypt'13. We show how to adapt and utilize the well-known half gates garbling technique (Zahur et al., Eurocrypt'15) to their constant round 2PFE scheme. Compared to their scheme, our resulting optimization significantly improves both underlying oblivious extended permutation (OEP) and secure 2-party computation (2PC) protocols, and yields a more than 40% reduction in overall communication cost. We next propose a novel and highly efficient 2PFE scheme based on the decisional Di e-Hellman (DDH) assumption. Our scheme consists of two protocols, one is utilized in the initial execution, and the other is in the subsequent runs. One of the novelties of our scheme over the state-of-the-art is that it results in a significant cost reduction when the same private function is evaluated more than once between the same or varying parties. To the best of our knowledge, this is the most efficient and the first 2PFE scheme that enjoys reusability feature. Our protocols achieve linear communication and computation complexities, and a constant number of rounds which is at most three (depending on the size of the inputs of the party that holds the function).
The area of "computing with encrypted data" has been studied by numerous authors in the past twenty years since it is fundamental to understanding properties of encryption and it has many practical applications. The related fundamental area of "secure function evaluation" has been studied since the mid 80's. In its basic two-party case, two parties (Alice and Bob) evaluate a known circuit over private inputs (or a private input and a private circuit). Much attention has been paid to the important issue of minimizing rounds of computation in this model. Namely, the number of communication rounds in which Alice and Bob need to engage in to evaluate a circuit on encrypted data securely. Advancements in these areas have been recognized as open problems and have remained open for a number of years. In this paper we give a one round, and thus round optimal, protocol for secure evaluation of circuits which is in polynomial time for NC/sup 1/ circuits. The protocol involves an input party sending encrypted input to a second party, a cryptocomputer, which evaluates the circuit (or a known circuit over its additional private input) non-interactively, securely and obliviously, and provides the output to the input party without learning it. This improves on previous (general) results that are specialized to the case of NC/sup 1/ circuits and require a constant number of communication rounds. We further suggest applications to network and mobile computing.
Cryptography grew to be much more than "the study of secret writing". Modern cryptography is concerned with establishing properties such as privacy, integrity and authenticity in protocols for secure communication and computation. This comes at a price: Cryptographic tools usually introduce an overhead, both in terms of communication complexity (that is, number and size of messages transmitted) and computational efficiency (that is, time and memory required). As in many settings communication between the parties involved is the bottleneck, this thesis is concerned with improving communication complexity in cryptographic protocols. One direction towards this goal is scalable cryptography: In many cryptographic schemes currently deployed, the security degrades linearly with the number of instances (e.g. encrypted messages) in the system. As this number can be huge in contexts like cloud computing, the parameters of the scheme have to be chosen considerably larger - and in particular depending on the expected number of instances in the system - to maintain security guarantees. We advance the state-of-the-art regarding scalable cryptography by constructing schemes where the security guarantees are independent of the number of instances. This allows to choose smaller parameters, even when the expected number of instances is immense. - We construct the first scalable encryption scheme with security against active adversaries which has both compact public keys and ciphertexts. In particular, we significantly reduce the size of the public key to only about 3% of the key-size of the previously most efficient scalable encryption scheme. (Gay,Hofheinz, and Kohl, CRYPTO, 2017) - We present a scalable structure-preserving signature scheme which improves both in terms of public-key and signature size compared to the previously best construction to about 40% and 56% of the sizes, respectively. (Gay, Hofheinz, Kohl, and Pan, EUROCRYPT, 2018) Another important area of cryptography is secure multi-party computation, where the goal is to jointly evaluate some function while keeping each party’s input private. In traditional approaches towards secure multi-party computation either the communication complexity scales linearly in the size of the function, or the computational efficiency is poor. To overcome this issue, Boyle, Gilboa, and Ishai (CRYPTO, 2016) introduced the notion of homomorphic secret sharing. Here, inputs are shared between parties such that each party does not learn anything about the input, and such that the parties can locally evaluate functions on the shares. Homomorphic secret sharing implies secure computation where the communication complexity only depends on the size of the inputs, which is typically much smaller than the size of the function. A different approach towards efficient secure computation is to split the protocol into an input-independent preprocessing phase, where long correlated strings are generated, and a very efficient online phase. One example for a useful correlation are authenticated Beaver triples, which allow to perform efficient multiplications in the online phase such that privacy of the inputs is preserved and parties deviating the protocol can be detected. The currently most efficient protocols implementing the preprocessing phase require communication linear in the number of triples to be generated. This results typically in high communication costs, as the online phase requires at least one authenticated Beaver triple per multiplication. We advance the state-of-the art regarding efficient protocols for secure computation with low communication complexity as follows. - We construct the first homomorphic secret sharing scheme for computing arbitrary functions in NC 1 (that is, functions that are computably by circuits with logarithmic depth) which supports message spaces of arbitrary size, has only negligible correctness error, and does not require expensive multiplication on ciphertexts. (Boyle, Kohl, and Scholl, EUROCRYPT, 2019) - We introduce the notion of a pseudorandom correlation generator for general correlations. Pseudorandom correlation generators allow to locally extend short correlated seeds into long pseudorandom correlated strings. We show that pseudorandom correlation generators can replace the preprocessing phase in many protocols, leading to a preprocessing phase with sublinear communication complexity. We show connections to homomorphic secret sharing schemes and give the first instantiation of pseudorandom correlation generators for authenticated Beaver triples at reasonable computational efficiency. (Boyle, Couteau, Gilboa, Ishai, Kohl, and Scholl, CRYPTO, 2019)
Abstract Private Simultaneous Messages (PSM) model is a minimal model for secure multiparty computation. Feige, Kilian, and Naor (STOC 1994) and Ishai (Cryptology and Information Security Series 2013) constructed PSM protocols based on quadratic residues. In this paper, we define QR-PSM protocols as a generalization of these protocols. A QR-PSM protocol is a PSM protocol whose decoding function outputs the quadratic residuosity modulo p of what is computed from messages. We design a QR-PSM protocol for any symmetric function f:{0,1}n→{0,1} f : { 0 , 1 } n → { 0 , 1 } of communication complexity O(n2) O ( n 2 ) . As far as we know, it is the most efficient PSM protocol for symmetric functions since the previously known best PSM protocol was of O(n2logn) O ( n 2 log n ) (Beimel et al., CRYPTO 2014). We also study the sizes of the underlying finite fields Fp F p in the protocols since the communication complexity of a QR-PSM protocol is proportional to the bit length of the prime p. We show that there is a prime p≤(1+o(1))N222N−2 p ≤ ( 1 + o ( 1 ) ) N 2 2 2 N - 2 such that any length-N pattern of quadratic (non)residues appears modulo p (and hence it can be used for general QR-PSM protocols), which improves the Peralta’s known result (Mathematics of Computation 1992) by a constant factor (1+2)2 ( 1 + 2 ) 2 .