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

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)

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, in which each player sends a random bit and the output is the majority of the players\u27 bits, can tolerate $t(n)=O (\sqrt n)$ even in the presence of \emph{adaptive} corruptions, and they conjectured that this is optimal for such adversaries. Lichtenstein, Linial, and Saks proved that the conjecture holds for protocols in which each player sends only a single bit. Their result has been the main progress on the conjecture during the last 30 years. In this work we revisit this question and ask: what about protocols where players can send longer messages? Can increased communication allow for a larger fraction of corrupt players? We introduce a model of \emph{strong adaptive} corruptions, in which an adversary sees all messages sent by honest parties in any given round and, based on the message content, decides whether to corrupt a party (and alter its message or sabotage its delivery) or not. This is in contrast to the (classical) adaptive adversary who can corrupt parties only based on past messages, and cannot alter messages already sent. We prove that any one-round coin-flipping protocol, \emph{regardless of message length}, can be secure against at most $\widetilde{O}(\sqrt n)$ 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. Going back to the standard adaptive model, we can now prove that any symmetric one-round protocol with arbitrarily long messages can tolerate at most $\widetilde{O}(\sqrt n)$ adaptive corruptions. At the heart of our results there is a novel use of the Minimax Theorem and a new technique for converting any one-round secure protocol with arbitrarily long messages into a secure one where each player sends only \polylog(n) bits. This technique may be of independent interest

On the Adaptive Security of the Threshold BLS Signature Scheme

Threshold signatures are a crucial tool for many distributed protocols. As shown by Cachin, Kursawe, and Shoup (PODC `00), schemes with unique signatures are of particular importance, as they allow to implement distributed coin flipping very efficiently and without any timing assumptions. This makes them an ideal building block for (inherently randomized) asynchronous consensus protocols. The threshold-BLS signature of Boldyreva (PKC `03) is both unique and very compact, but unfortunately lacks a security proof against adaptive adversaries. Thus, current consensus protocols either rely on less efficient alternatives or are not adaptively secure. In this work, we revisit the security of the threshold BLS signature by showing the following results, assuming t adaptive corruptions: - We give a modular security proof that follows a two-step approach: 1) We introduce a new security notion for distributed key generation protocols (DKG). We show that it is satisfied by several protocols that previously only had a static security proof. 2) Assuming any DKG protocol with this property, we then prove unforgeability of the threshold BLS scheme. Our reductions are tight and can be used to substantiate real-world parameter choices. - To justify our use of strong assumptions such as the algebraic group model (AGM) and the hardness of one-more-discrete logarithm (OMDL), we prove an impossibility result: Even in the AGM, a strong interactive assumption is required in order to prove the scheme secure

Cryptography

The Oberwolfach workshop Cryptography brought together scientists from cryptography with mathematicians specializing in the algorithmic problems underlying cryptographic security. The goal of the workshop was to stimulate interaction and collaboration that enables a holistic approach to designing cryptography from the mathematical foundations to practical applications. The workshop covered basic computational problems such as factoring and computing discrete logarithms and short vectors. It addressed fundamental research results leading to innovative cryptography for protecting security and privacy in cloud applications. It also covered some practical applications

On Black-Box Complexity and Adaptive, Universal Composability of Cryptographic Tasks

Two main goals of modern cryptography are to identify the minimal assumptions necessary to construct secure cryptographic primitives as well as to construct secure protocols in strong and realistic adversarial models. In this thesis, we address both of these fundamental questions. In the first part of this thesis, we present results on the black-box complexity of two basic cryptographic primitives: non-malleable encryption and optimally-fair coin tossing. Black-box reductions are reductions in which both the underlying primitive as well as the adversary are accessed only in an input-output (or black-box) manner. Most known cryptographic reductions are black-box. Moreover, black-box reductions are typically more efficient than non-black-box reductions. Thus, the black-box complexity of cryptographic primitives is a meaningful and important area of study which allows us to gain insight into the primitive. We study the black box complexity of non-malleable encryption and optimally-fair coin tossing, showing a positive result for the former and a negative one for the latter. Non-malleable encryption is a strong security notion for public-key encryption, guaranteeing that it is impossible to "maul" a ciphertext of a message m into a ciphertext of a related message. This security guarantee is essential for many applications such as auctions. We show how to transform, in a black-box manner, any public-key encryption scheme satisfying a weak form of security, semantic security, to a scheme satisfying non-malleability. Coin tossing is perhaps the most basic cryptographic primitive, allowing two distrustful parties to flip a coin whose outcome is 0 or 1 with probability 1/2. A fair coin tossing protocol is one in which the outputted bit is unbiased, even in the case where one of the parties may abort early. However, in the setting where parties may abort early, there is always a strategy for one of the parties to impose bias of Omega(1/r) in an r-round protocol. Thus, achieving bias of O(1/r) in r rounds is optimal, and it was recently shown that optimally-fair coin tossing can be achieved via a black-box reduction to oblivious transfer. We show that it cannot be achieved via a black-box reduction to one-way function, unless the number of rounds is at least Omega(n/log n), where n is the input/output length of the one-way function. In the second part of this thesis, we present protocols for multiparty computation (MPC) in the Universal Composability (UC) model that are secure against malicious, adaptive adversaries. In the standard model, security is only guaranteed in a stand-alone setting; however, nothing is guaranteed when multiple protocols are arbitrarily composed. In contrast, the UC model, introduced by (Canetti, 2000), considers the execution of an unbounded number of concurrent protocols, in an arbitrary, and adversarially controlled network environment. Another drawback of the standard model is that the adversary must decide which parties to corrupt before the execution of the protocol commences. A more realistic model allows the adversary to adaptively choose which parties to corrupt based on its evolving view during the protocol. In our work we consider the the adaptive UC model, which combines these two security requirements by allowing both arbitrary composition of protocols and adaptive corruption of parties. In our first result, we introduce an improved, efficient construction of non-committing encryption (NCE) with optimal round complexity, from a weaker primitive we introduce called trapdoor-simulatable public key encryption (PKE). NCE is a basic primitive necessary to construct protocols secure under adaptive corruptions and in particular, is used to construct oblivious transfer (OT) protocols secure against semi-honest, adaptive adversaries. Additionally, we show how to realize trapdoor-simulatable PKE from hardness of factoring Blum integers, thus achieving the first construction of NCE from hardness of factoring. In our second result, we present a compiler for transforming an OT protocol secure against a semi-honest, adaptive adversary into one that is secure against a malicious, adaptive adversary. Our compiler achieves security in the UC model, assuming access to an ideal commitment functionality, and improves over previous work achieving the same security guarantee in two ways: it uses black-box access to the underlying protocol and achieves a constant multiplicative overhead in the round complexity. Combining our two results with the work of (Ishai et al., 2008), we obtain the first black-box construction of UC and adaptively secure MPC from trapdoor-simulatable PKE and the ideal commitment functionality

Optimally-secure Coin-tossing against a Byzantine Adversary

In their seminal work, Ben-Or and Linial (1985) introduced the full information model for collective coin-tossing protocols involving $n$ processors with unbounded computational power using a common broadcast channel for all their communications. The design and analysis of coin-tossing protocols in the full information model have close connections to diverse fields like extremal graph theory, randomness extraction, cryptographic protocol design, game theory, distributed protocols, and learning theory. Several works have focused on studying the asymptotically best attacks and optimal coin-tossing protocols in various adversarial settings. While one knows the characterization of the exact or asymptotically optimal protocols in some adversarial settings, for most adversarial settings, the optimal protocol characterization remains open. For the cases where the asymptotically optimal constructions are known, the exact constants or poly-logarithmic multiplicative factors involved are not entirely well-understood. In this work, we study $n$-processor coin-tossing protocols where every processor broadcasts an arbitrary-length message once. Note that, in this setting, which processor speaks and its message distribution may depend on the messages broadcast so far. An adaptive Byzantine adversary, based on the messages broadcast so far, can corrupt $k=1$ processor. A bias-$X$ coin-tossing protocol outputs 1 with probability $X$; 0 with probability $(1-X)$. For a coin-tossing protocol, its insecurity is the maximum change in the output distribution (in the statistical distance) that an adversarial strategy can cause. Our objective is to identify optimal bias-$X$ coin-tossing protocols with minimum insecurity, for every $X\in[0,1]$. Lichtenstein, Linial, and Saks (1989) studied bias-$X$ coin-tossing protocols in this adversarial model under the highly restrictive constraint that each party broadcasts an independent and uniformly random bit. The underlying message space is a well-behaved product space, and $X\in[0,1]$ can only be integer multiples of $1/2^n$, which is a discrete problem. The case where every processor broadcasts only an independent random bit admits simplifications, for example, the collective coin-tossing protocol must be monotone. Surprisingly, for this class of coin-tossing protocols, the objective of reducing an adversary’s ability to increase the expected output is equivalent to reducing an adversary’s ability to decrease the expected output. Building on these observations, Lichtenstein, Linial, and Saks proved that the threshold coin-tossing protocols are optimal for all $n$ and $k$. In a sequence of works, Goldwasser, Kalai, and Park (2015), Kalai, Komargodski, and Raz (2018), and (independent of our work) Haitner and Karidi-Heller (2020) prove that k=\mathcal{O}\left(\sqrt n\cdot \polylog{n}\right) corruptions suffice to fix the output of any bias-X coin-tossing protocol. These results consider parties who send arbitrary-length messages, and each processor has multiple turns to reveal its entire message. However, optimal protocols robust to a large number of corruptions do not have any apriori relation to the optimal protocol robust to $k=1$ corruption. Furthermore, to make an informed choice of employing a coin-tossing protocol in practice, for a fixed target tolerance of insecurity, one needs a precise characterization of the minimum insecurity achieved by these coin-tossing protocols. We rely on an inductive approach to constructing coin-tossing protocols to study a proxy potential function measuring the susceptibility of any bias-$X$ coin-tossing protocol to attacks in our adversarial model. Our technique is inherently constructive and yields protocols that minimize the potential function. It happens to be the case that threshold protocols minimize the potential function. We demonstrate that the insecurity of these threshold protocols is 2-approximate of the optimal protocol in our adversarial model. For any other $X\in[0,1]$ that threshold protocols cannot realize, we prove that an appropriate (convex) combination of the threshold protocols is a 4-approximation of the optimal protocol