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Stronger Security for Reusable Garbled Circuits, General Definitions and Attacks

Shweta Agrawal

Published Jan 1, 2016• 2 citations
OpenAlex

Abstract

We construct a functional encryption scheme for circuits which simultaneously achieves and improves upon the security of the current best known, and incomparable, constructions from standard assumptions: reusable garbled circuits by Goldwasser, Kalai, Popa, Vaikuntanathan and Zeldovich (STOC 2013) [GKP+13] and predicate encryption for circuits by Gorbunov, Vaikuntanathan and Wee (CRYPTO 2015) [GVW15]. Our scheme is secure based on the learning with errors (LWE) assumption. Our construction implies: 1. A new construction for reusable garbled circuits that achieves stronger security than the only known prior construction [GKP+13]. 2. A new construction for bounded collusion functional encryption with substantial efficiency benefits: our public parameters and ciphertext size incur an additive growth of \(O(Q^2)\), where Q is the number of permissible queries (We note that due to a lower bound [AGVW13], the ciphertext size must necessarily grow with Q). Additionally, the ciphertext of our scheme is succinct, in that it does not depend on the size of the circuit. By contrast, the prior best construction [GKP+13, GVW12] incurred a multiplicative blowup of \(O(Q^4)\) in both the public parameters and ciphertext size. However, our scheme is secure in a weaker game than [GVW12].

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•Jan 1, 2014•4 citations
We construct a functional encryption scheme for circuits which simultaneously achieves and improves upon the security of the current best known, and incomparable, constructions from standard assumptions: reusable garbled circuits by Goldwasser, Kalai, Popa, Vaikuntanathan and Zeldovich (STOC 2013) [GKP+13] and predicate encryption for circuits by Gorbunov, Vaikuntanathan and Wee (CRYPTO 2015) [GVW15]. Our scheme is secure based on the learning with errors (LWE) assumption. Our construction implies:
A new construction for reusable garbled circuits that achieves stronger security than the only known prior construction [GKP+13].
A new construction for bounded collusion functional encryption with substantial efficiency benefits: our public parameters and ciphertext size incur an additive growth of (O(Q^2)), where Q is the number of permissible queries (We note that due to a lower bound [AGVW13], the ciphertext size must necessarily grow with Q). Additionally, the ciphertext of our scheme is succinct, in that it does not depend on the size of the circuit. By contrast, the prior best construction [GKP+13, GVW12] incurred a multiplicative blowup of (O(Q^4)) in both the public parameters and ciphertext size. However, our scheme is secure in a weaker game than [GVW12].
Functional encryption is a powerful primitive: given an encryption Enc(x) of a value x and a secret key skf corresponding to a circuit f, it enables efficient computation of f(x) without revealing any additional information about x. Constructing functional encryption schemes with succinct ciphertexts that guarantee security for even a single secret key (for a general function f) is an important open problem with far reaching applications, which this paper addresses. Our main result is a functional encryption scheme for any general function f of depth d, with succinct ciphertexts whose size grows with the depth d rather than the size of the circuit for f. We prove the security of our construction based on the intractability of the learning with error (LWE) problem. More generally, we show how to construct a functional encryption scheme from any public-index predicate encryption scheme and fully homomorphic encryption scheme. Previously, the only known constructions of functional encryption were either for specific inner product predicates, or for a weak form of functional encryption where the ciphertext size grows with the size of the circuit for f. We demonstrate the power of this result, by using it to construct a reusable circuit garbling scheme with
Garbled circuits, introduced by Yao in the mid 80s, allow computing a function f on an input x without leaking anything about f or x besides f(x). Garbled circuits found numerous applications, but every known construction suffers from one limitation: it offers no security if used on multiple inputs x. In this paper, we construct for the first time reusable garbled circuits. The key building block is a new succinct single-key functional encryption scheme. Functional encryption is an ambitious primitive: given an encryp-tion Enc(x) of a value x, and a secret key skf for a function f, anyone can compute f(x) without learning any other information about x. We construct, for the first time, a succinct functional encryption scheme for any polynomial-time function f where succinctness means that the ciphertext size does not grow with the size of the circuit for f, but only with its depth. The security of our construction is based on the intractability of the Learning with Errors (LWE) problem and holds as long as an adversary has access to a single key skf (or even an a priori bounded number of keys for different functions). Building on our succinct single-key functional encryption scheme, we show several new applications in addition to reusable garbled circuits, such as a paradigm for general function obfuscation which we call token-based obfuscation, homomorphic encryption for a class of Turing machines where the evaluation runs in input-specific time rather than worst-case time, and a scheme for delegating computation which is publicly verifiable and maintains the privacy of the computation.
We provide a single-key functional encryption scheme for Deterministic Finite Automata (DFA). The secret key of our scheme is associated with a DFA M, and a ciphertext is associated with an input x of arbitrary length. The decryptor learns M(x) and nothing else. The ciphertext and key sizes achieved by our scheme are optimal – the size of the public parameters is independent of the size of the machine or data being encrypted, the secret key size depends only on the machine size and the ciphertext size depends only on the input size. Our scheme achieves full functional encryption in the “private index model”, namely the entire input x is hidden (as against x being public and a single bit b being hidden). Our single key FE scheme can be compiled with symmetric key encryption to yield reusable garbled DFAs for arbitrary size inputs, that achieves machine and input privacy along with reusability under a strong simulation based definition of security. We generalize this to a functional encryption scheme for Turing machines TMFE which has short public parameters that are independent of the size of the machine or the data being encrypted, short function keys, and input-specific decryption time. However, the ciphertext of our construction is large and depends on the worst case running time of the Turing machine (but not its description size). These provide the first FE schemes that support unbounded length inputs, allow succinct public and function keys and rely on LWE. Our construction relies on a new and arguably natural notion of decomposable functional encryption which may be of independent interest.
We construct a general-purpose multi-input functional encryption scheme in the private-key setting. Namely, we construct a scheme where a functional key corresponding to a function f enables a user holding encryptions of x1; : : : ; xt to compute f(x1; : : : ; xt) but nothing else. This is achieved starting from any general-purpose private-key single-input scheme (without any additional assumptions), and is proven to be adaptively secure for any constant number of inputs t. Moreover, it can be extended to a super-constant number of inputs assuming that the underlying single-input scheme is sub-exponentially secure. Instantiating our construction with existing single-input schemes, we obtain multi-input schemes that are based on a variety of assumptions (such as indistinguishability obfuscation, multilinear maps, learning with errors, and even one-way functions), offering various trade-offs between security and efficiency. Previous and concurrent constructions of multi-input functional encryption schemes either rely on stronger assumptions and provided weaker security guarantees (Goldwasser et al. [EU-ROCRYPT '14], and Ananth and Jain [CRYPTO '15]), or relied on multilinear maps and could
Recently, privacy has become one of the hottest topics in the world of the Internet-based communications and computation. Many applications, such as cloud computing and various cloud services, require information security and privacy. However, in many cases the level of security in current solutions is insufficient. Methods to ensure secure multiparty computation have been studied for decades, but because of the needs of the modern computation this topic has recently become increasingly popular. Yao's garbled circuit is one method to compute a function f on an argument x privately without revealing any information about f and x except f(x). Unfortunately, current constructions for garbled circuits have limitations - either efficient construction at the cost of reusability or reusability at the cost of efficiency. In this paper we aim at somewhere between these two extremes by defining new security measures for garbling schemes. These definitions support design of reusable garbled circuits that achieve adaptive security even when the function argument is handled bitwise. Moreover, we show how the new classes are related to the existing security classes of garbling schemes. As a result, we obtain an extended hierarchy for the classes of garbling schemes.