Adaptively Secure Coin-Flipping, Revisited

The full-information model was introduced by Ben-Or and Linial in 1985 to study collective coin-flipping: the problem of generating a common bounded-bias bit in a network of $n$ players with $t=t(n)$ faults. They showed that the majority protocol can tolerate $t=O(\sqrt n)$ adaptive corruptions, and conjectured that this is optimal in the adaptive setting. Lichtenstein, Linial, and Saks proved that the conjecture holds for protocols in which each player sends a single bit. Their result has been the main progress on the conjecture in the last 30 years. In this work we revisit this question and ask: what about protocols involving longer messages? Can increased communication allow for a larger fraction of faulty players? We introduce a model of strong adaptive corruptions, where in each round, the adversary sees all messages sent by honest parties and, based on the message content, decides whether to corrupt a party (and intercept his message) or not. We prove that any one-round coin-flipping protocol, regardless of message length, is secure against at most $\tilde{O}(\sqrt n)$ strong adaptive corruptions. Thus, increased message length does not help in this setting. We then shed light on the connection between adaptive and strongly adaptive adversaries, by proving that for any symmetric one-round coin-flipping protocol secure against $t$ adaptive corruptions, there is a symmetric one-round coin-flipping protocol secure against $t$ strongly adaptive corruptions. Returning to the standard adaptive model, we can now prove that any symmetric one-round protocol with arbitrarily long messages can tolerate at most $\tilde{O}(\sqrt n)$ adaptive corruptions. At the heart of our results lies a novel use of the Minimax Theorem and a new technique for converting any one-round secure protocol into a protocol with messages of $polylog(n)$ bits. This technique may be of independent interest

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Attacks on the Fiat-Shamir paradigm and program obfuscation

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2006.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 115-119).The goal of cryptography is to construct *secure* and *efficient* protocols for various tasks. Unfortunately, it is often the case that protocols that are provably secure are not efficient enough for practical use. As a result, most protocols used in practice are *heuristics* that lack proofs of security. These heuristics are typically very efficient and are believed to be secure, though no proof of security has been provided. In this thesis we study the security of some of these popular heuristics. In particular, we focus on two types of heuristics: (1) the Fiat-Shamir heuristic for constructing digital signature schemes, and (2) heuristics for obfuscation. We show that, in some sense, both of these types of heuristics are insecure. Thus, this thesis consists of two parts: (1) The insecuirty of the Fiat-Shamir paradigm: The Fiat-Shamir heuristic provides a general method for transforming secure 3-round public-coin identification schemes into digital signature schemes. The idea of the transformation is to replace the random (second-round) message of the verifier in the identification scheme, with the value of some deterministic hash function evaluated on the first-round message (sent by the prover) and on the message to be signed.(cont.) The Fiat-Shamir methodology for producing digital signature schemes quickly gained popularity both in theory and in practice, as it yields efficient and easy to implement digital signature schemes. The most important question however remained open: are the digital signature schemes produced by the Fiat-Shamir methodology secure? In this thesis, we answer this question negatively. We show that there exist secure 3-round public-coin identification schemes for which the Fiat-Shamir transformation yields *insecure* digital signature schemes for *any* hash function used by the transformation. This is in contrast to the work of Pointcheval and Stern, who proved that the Fiat-Shamir methodology always produces digital signature schemes that are secure against chosen message attacks in the ``Random Oracle Model" -- when the hash function is modeled by a random oracle. (2) The impossibility of obfuscation: The goal of code obfuscation is to make a program completely "unintelligible" while preserving its functionality. Obfuscation has been used for many years in attempts to prevent reverse engineering, e.g ., in copy protection, licensing schemes, and games.(cont.) As a result, many heuristics for obfuscation have emerged, and the important question that remained is: are these heuristics for obfuscation secure? In this thesis, we show that there are many "natural" classes of functions for which obfuscation is not at all possible. This impossibility result holds in an augmentation of the formal obfuscation model of Barak, et al. (2001) that includes auxiliary input. In both of these parts, among other tools, we make new usage of Barak's technique for taking advantage of non black-box access to a program, this time in the context of digital signature schemes and in the context of obfuscation.by Yael Tauman Kalai.Ph.D

A Lower Bound for Adaptively-Secure Collective Coin-Flipping Protocols

In 1985, Ben-Or and Linial (Advances in Computing Research \u2789) introduced the collective coin-flipping problem, where n parties communicate via a single broadcast channel and wish to generate a common random bit in the presence of adaptive Byzantine corruptions. In this model, the adversary can decide to corrupt a party in the course of the protocol as a function of the messages seen so far. They showed that the majority protocol, in which each player sends a random bit and the output is the majority value, tolerates O(sqrt n) adaptive corruptions. They conjectured that this is optimal for such adversaries. We prove that the majority protocol is optimal (up to a poly-logarithmic factor) among all protocols in which each party sends a single, possibly long, message. Previously, such a lower bound was known for protocols in which parties are allowed to send only a single bit (Lichtenstein, Linial, and Saks, Combinatorica \u2789), or for symmetric protocols (Goldwasser, Kalai, and Park, ICALP \u2715)

Proofs of Ignorance and Applications to 2-Message Witness Hiding

We consider the following paradoxical question: Can one prove lack of knowledge? We define the notion of \u27Proofs of Ignorance\u27, construct such proofs, and use these proofs to construct a 2-message witness hiding protocol for all of NP. More specifically, we define a proof of ignorance (PoI) with respect to any language L in NP and distribution D over instances in L. Loosely speaking, such a proof system allows a prover to generate an instance x according to D along with a proof that she does not know a witness corresponding to x. We construct construct a PoI protocol for any random self-reducible NP language L that is hard on average. Our PoI protocol is non-interactive assuming the existence of a common reference string. We use such a PoI protocol to construct a 2-message witness hiding protocol for NP with adaptive soundness. Constructing a 2-message WH protocol for all of NP has been a long standing open problem. We construct our witness hiding protocol using the following ingredients (where T is any super-polynomial function in the security parameter): 1. T-secure PoI protocol, 2. T-secure non-interactive witness indistinguishable (NIWI) proofs, 3. T-secure rerandomizable encryption with strong KDM security with bounded auxiliary input, where the first two ingredients can be constructed based on the $T$-security of DLIN. At the heart of our witness-hiding proof is a new non-black-box technique. As opposed to previous works, we do not apply an efficiently computable function to the code of the cheating verifier, rather we resort to a form of case analysis and show that the prover\u27s message can be simulated in both cases, without knowing in which case we reside