We show that the most common flavors of noisy leakage can be simulated in the information-theoretic setting using a single query of bounded leakage, up to a small statistical simulation error and a slight loss in the leakage parameter. The latter holds true in particular for one of the most used noisy-leakage models, where the noisiness is measured using the conditional average min-entropy (Naor and Segev, CRYPTO’09 and SICOMP’12). Our reductions between noisy and bounded leakage are achieved in two steps. First, we put forward a new leakage model (dubbed the dense leakage model) and prove that dense leakage can be simulated in the information-theoretic setting using a single query of bounded leakage, up to small statistical distance. Second, we show that the most common noisy-leakage models fall within the class of dense leakage, with good parameters. Third, we prove lower bounds on the amount of bounded leakage required for simulation with sub-constant error, showing that our reductions are nearly optimal. In particular, our results imply that useful general simulation of noisy leakage based on statistical distance and mutual information is impossible. We also provide a complete picture of the relationships between different noisy-leakage models. Our result finds applications to leakage-resilient cryptography, where we are often able to lift security in the presence of bounded leakage to security in the presence of noisy leakage, both in the information-theoretic and in the computational setting. Remarkably, this lifting procedure makes only black-box use of the underlying schemes. Additionally, we show how to use lower bounds in communication complexity to prove that bounded-collusion protocols (Kumar, Meka, and Sahai, FOCS’19) for certain functions do not only require long transcripts, but also necessarily need to reveal enough information about the inputs.
We show that the most common flavors of noisy leakage can be simulated in the information-theoretic setting using a single query of bounded leakage, up to a small statistical simulation error and a slight loss in the leakage parameter. The latter holds true in particular for one of the most used noisy-leakage models, where the noisiness is measured using the conditional average min-entropy (Naor and Segev, CRYPTO’09 and SICOMP’12). Our reductions between noisy and bounded leakage are achieved in two steps. First, we put forward a new leakage model (dubbed the dense leakage model) and prove that dense leakage can be simulated in the information-theoretic setting using a single query of bounded leakage, up to small statistical distance. Second, we show that the most common noisy-leakage models fall within the class of dense leakage, with good parameters. Third, we prove lower bounds on the amount of bounded leakage required for simulation with sub-constant error, showing that our reductions are nearly optimal. In particular, our results imply that useful general simulation of noisy leakage based on statistical distance and mutual information is impossible. We also provide a complete picture of the relationships between different noisy-leakage models. Our result finds applications to leakage-resilient cryptography, where we are often able to lift security in the presence of bounded leakage to security in the presence of noisy leakage, both in the information-theoretic and in the computational setting. Remarkably, this lifting procedure makes only black-box use of the underlying schemes. Additionally, we show how to use lower bounds in communication complexity to prove that bounded-collusion protocols (Kumar, Meka, and Sahai, FOCS’19) for certain functions do not only require long transcripts, but also necessarily need to reveal enough information about the inputs.
In this work, we show how to use the positive results on succinct argument systems to prove impossibility results on leakage-resilient black-box zero knowledge. This recently proposed notion of zero knowledge deals with an adversary that can make leakage queries on the state of the prover. Our result holds for black-box simulation only and we also give some insights on the non-black-box case. Additionally, we show that, for several functionalities, leakage-resilient multi-party computation is impossible (regardless of the number of players and even if just one player is corrupted). More in details, we achieve the above results by extending a technique of [Nielsen, Venturi, Zottarel – PKC 13] to prove lower bounds for leakage-resilient security. Indeed, we use leakage queries to run an execution of a communication-efficient protocol in the head of the adversary. Moreover, to defeat the black-box simulator we connect the above technique for leakage resilience to security against reset attacks. Our results show that the open problem of [Ananth, Goyal, Pandey – Crypto 14] (i.e., continual leakage-resilient proofs without a common reference string) has a negative answer when security through black-box simulation is desired. Moreover our results close the open problem of [Boyle et al. – STOC 12] for the case of black-box simulation (i.e., the possibility of continual leakage-resilient secure computation without a leak-free interactive preprocessing).
A long series of works have established far reaching impossibility results for concurrently secure computation. On the other hand, some positive results have also been obtained according to various weaker notions of security (such as by using a super-polynomial time simulator). This suggest that somehow, “not all is lost in the concurrent setting.” In this work, we ask what and exactly how much private information can an adversary learn by launching a concurrent attack? Inspired by the recent works on leakage-resilient protocols, we consider a security model where the ideal world adversary (a.k.a simulator) is allowed to query the trusted party for some “leakage ” on the honest party inputs. Intuitively, the amount of leakage required by the simulator upper bounds the security loss in the real world. This model generalizes the multiple ideal query model (MIQ) for concurrent security proposed by Goyal, Jain and Ostrovsky [CRYPTO’10]. Positive Results. As our main result, we show, for the first time, how to compute any efficiently computable functionality in the concurrent setting such that full security is guaranteed for “most ” of the sessions. At the heart of this result is a new precise simulation strategy (first
In this work, we consider the natural goal of designing secret sharing schemes that ensure security against an adversary who may learn some “leaked'' information about all the shares. We say that a secret sharing scheme is p-party leakage-resilient, if the secret remains statistically hidden even after a computationally unbounded adversary learns a bounded amount of leakage, where each bit of leakage adaptively and jointly depends on the shares of an adaptively chosen subset of p parties. Existing multi-party secret sharing schemes (Dziembowski and Pietrzak FOCS 07), (Goyal and Kumar STOC 18) and (Benhamouda, Degwekar, Ishai and Rabin CRYPTO 18) have focused on handling non-adaptive and individual leakage for (limited special cases of) threshold secret sharing schemes. (1) We give an unconditional compiler that transforms any secret sharing scheme on n parties into a p-party leakage-resilient one for p upto O(log n). This yields the first multi-party secret sharing schemes that are secure against adaptive or joint leakage. (2) As a natural extension, we initiate the study of leakage-resilient non-malleable secret sharing. We empower the adversary to adaptively leak from each of the shares and then use the leakage to tamper with all of them arbitrarily and independently. Leveraging our p-party leakage-resilient schemes, we compile any secret sharing scheme into a non-malleable one ensuring that any such tampering either preserves the secret or completely `destroys' it. This improves upon the non-malleable secret sharing scheme of (Goyal and Kumar CRYPTO 18) where no leakage was permitted. Leakage-resilient non-malleable codes can be seen as 2-out-of-2 schemes satisfying our guarantee and have already found many applications in cryptography. (3) Our constructions rely on a clean connection we draw to communication complexity in the well-studied number-on-forehead (NOF) model and rely on functions that have strong communication-complexity lower bounds in the NOF model (in a black-box way). We get efficient p-party leakage-resilient schemes for p upto O(log n) as our share sizes have exponential dependence on p. We observe that improving this exponential dependence, even for simultaneous, non-adaptive leakage, will lead to progress on longstanding open problems in complexity theory.
Secret sharing is an instrumental tool for sharing secret keys in distributed systems. In a classical threshold setting, this involves a dealer who has a secret/key, a set of parties/users to which shares of the secret are sent, and a threshold on the number of users whose presence is needed in order to recover the secret. In secret sharing, secure links with no leakage are often assumed between the involved parties. However, when the users are nodes in a communication network and all the links are physical links, e.g., wireless, such assumptions are not valid anymore. In order to study this critical problem, we propose a statistical leakage model of secret sharing, where some noisy versions of all the secret shares might be independently leaked to an adversary. We then study the resilience of the seminal Shamir's secret sharing scheme with statistical leakage, and bound certain measures of security (i.e., semantic security, mutual information security), given other parameters of the system including the amount of leakage from each secret share. We show that for an extreme scenario of Shamir's scheme, in particular when the underlying field characteristic is 2, the security of each bit of the secret against leakage improves exponentially with the number of users. To the best of our knowledge, this is the first attempt towards understanding secret sharing under general statistical noisy leakage.
In [AGP14] Ananth et al. showed that continual leakage-resilient non-transferable interactive proofs exist when a leak-free input-encoding phase is allowed and a common reference string is available. They left open the problem of removing the need of a common reference string. In [BGJK12] Boyle et al. showed that for some interesting functionalities continual leakage-resilient secure computation is possible when leak-free interactive preprocessing and input-encoding phases are allowed. They left open the problem of removing the interactive pre-processing. In this work we study the above questions. Our main contribution shows that leakage-resilient black-box zero-knowledge is impossible when relying on a leak-free input-encoding phase only (i.e., without CRS/preprocessing). Additionally, we also show that leakage-resilient multi-party computation for all functionalities is impossible (regardless of the number of players assuming just one corrupted player) when relying only on a leak-free input-encoding phase (i.e., without CRS/preprocessing). Our results are achieved by extending a technique of [NVZ13] to prove lower bounds for leakage-resilient security. Indeed as in [NVZ13] we use leakage queries to run an execution of a communication-efficient protocol in the head of the adversary. Moreover, to defeat the black-box simulator we connect the above technique for leakage resilience to security against reset attacks.
Given a sequence of N independent sources X1,X2,…,XN∼{0,1}n, how many of them must be good (i.e., contain some min-entropy) in order to extract a uniformly random string? This question was first raised by Chattopadhyay, Goodman, Goyal and Li (STOC '20), motivated by applications in cryptography, distributed computing, and the unreliable nature of real-world sources of randomness. In their paper, they showed how to construct explicit low-error extractors for just K≥N1/2 good sources of polylogarithmic min-entropy. In a follow-up, Chattopadhyay and Goodman improved the number of good sources required to just K≥N0.01 (FOCS '21). In this paper, we finally achieve K=3. Our key ingredient is a near-optimal explicit construction of a new pseudorandom primitive, called a leakage-resilient extractor (LRE) against number-on-forehead (NOF) protocols. Our LRE can be viewed as a significantly more robust version of Li's low-error three-source extractor (FOCS '15), and resolves an open question put forth by Kumar, Meka, and Sahai (FOCS '19) and Chattopadhyay, Goodman, Goyal, Kumar, Li, Meka, and Zuckerman (FOCS '20). Our LRE construction is based on a simple new connection we discover between multiparty communication complexity and non-malleable extractors, which shows that such extractors exhibit strong average-case lower bounds against NOF protocols.