Ashutosh Kumar 0002

FOCS Conference 2020 Conference Paper

Extractors and Secret Sharing Against Bounded Collusion Protocols

• Eshan Chattopadhyay
• Jesse Goodman
• Vipul Goyal
• Ashutosh Kumar 0002
• Xin Li 0006
• Raghu Meka
• David Zuckerman

In a recent work, Kumar, Meka, and Sahai (FOCS 2019) introduced the notion of bounded collusion protocols (BCPs). BCPs are multiparty communication protocols in which N parties, holding n bits each, attempt to compute some joint function of their inputs, f: ({0, 1} n ) N →{0, 1}. In each round, p parties (the collusion bound) work together to write a single bit on a public blackboard, and the protocol continues until every party knows the value of f. BCPs are a natural generalization of the well-studied number-in-hand (NIH) and number-on-forehead (NOF) models, which are just endpoints on this rich spectrum of protocols (corresponding to p=1 and p=N-1, respectively). In this work, we investigate BCPs more thoroughly, and answer questions about them in the context of communication complexity, randomness extractors, and secret sharing. 1. First, we provide explicit lower bounds against BCPs. Our lower bounds offer a tradeoff between collusion and complexity, and are of the form n Ω(1) when p=0. 99N parties collude. This bound is independent of the relationship between N, n, whereas all previous bounds became trivial when. 2. Second, we provide explicit leakage-resilient extractors against BCPs. Also known as cylinder-intersection extractors, these objects are multi-source extractors of the form Ext: ({0, 1} n ) N →{0, 1}, whose output looks uniform even conditioned on the bits produced (“leaked”) by a BCP executed over the inputs of the extractor. Our extractors work for sources with min-entropy k ≥ polylog(n) against BCPs with collusion p ≤ N-2. Previously, all such extractors required min-entropy k ≥ 0. 99n even when p ≤ O(1). 3. Third, we provide efficient leakage-resilient secret sharing schemes against BCPs. These cryptographic primitives are standard t-out-of- N secret sharing schemes, equipped with an additional guarantee that the secret remains hidden even if the individuals participate in a BCP using their shares. Our schemes can handle collusion up to p ≤ O(t/logt), whereas the previous best scheme required p ≤ O(logN). Along the way, we also construct objects that are more general than those listed above (i. e. , compilers), objects that are more specialized (and stronger) than those listed above, and resolve open questions posed by Goyal and Kumar (STOC 2018) and Kumar, Meka, and Sahai (FOCS 2019).

FOCS Conference 2019 Conference Paper

Leakage-Resilient Secret Sharing Against Colluding Parties

• Ashutosh Kumar 0002
• Raghu Meka
• Amit Sahai

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.

STOC Conference 2018 Conference Paper

Non-malleable secret sharing

• Vipul Goyal
• Ashutosh Kumar 0002