International Association
for Cryptologic Research

Policy-compliant signatures (PCS) are a recently introduced primitive by Badertscher et al. [TCC 2021] in which a central authority distributes secret and public keys associated with sets of attributes (e.g., nationality, affiliation with a specific department, or age) to its users. The authority also enforces a policy determining which senders can sign messages for which receivers based on a joint check of their attributes. For example, senders and receivers must have the same nationality, or only senders that are at least 18 years old can send to members of the computer science department. PCS further requires attribute-privacy -- nothing about the users' attributes is revealed from their public keys and signatures apart from whether the attributes satisfy the policy or not. The policy in a PCS scheme is fixed once and for all during the setup. Therefore, a policy update requires a redistribution of all keys. This severely limits the practicality of PCS. In this work, we introduce the notion of updatable policy-compliant signatures (UPCS) extending PCS with a mechanism to efficiently update the policy without redistributing keys to all participants. We define the notion of UPCS and provide the corresponding security definitions. We then provide a generic construction of UPCS based on digital signatures, a NIZK proof system, and a so-called secret-key two-input partially-hiding predicate encryption (2-PHPE) scheme. Unfortunately, the only known way to build the latter for general two-input predicates is using indistinguishability obfuscation. We show that the reliance on the heavy tool of 2-PHPE is inherent to build UPCS by proving that non-interactive UPCS implies 2-PHPE. To circumvent the reliance on 2-PHPE, we consider interactive UPCS, which allows sender and receiver to interact during the message signing. In this setting, we present two UPCS schemes: the first one requires only a digital signature scheme, a NIZK proof system, and secure two-party computation. This scheme works for arbitrary policies, but requires senders and receivers to engage in the two-party computation for each policy update. Our second scheme additionally requires a (single-input) predicate-encryption scheme and only requires the sender and receiver to interact ones independent of the updates. In contrast to 2-PHPE, single-input predicate encryption supporting certain predicate classes are known to exist (e.g., from pairings) under more concrete and well-understood assumptions.

Messages in large-scale networks such as blockchain systems are typically disseminated using flooding protocols, in which parties send the message to a random set of peers until it reaches all parties. Optimizing the communication complexity of such protocols and, in particular, the per-party communication complexity is of primary interest since nodes in a network are often subject to bandwidth constraints. Previous flooding protocols incur a per-party communication complexity of $\Omega(l\cdot \gamma^{-1} \cdot (\log(n) + \kappa))$ bits to disseminate an $l$-bit message among $n$ parties with security parameter~$\kappa$ when it is guaranteed that a $\gamma$ fraction of the parties remain honest. In this work, we present the first flooding protocols with a per-party communication complexity of $O(l\cdot \gamma^{-1})$ bits. We further show that this is asymptotically optimal and that our protocols can be instantiated provably securely in the usual setting for proof-of-stake blockchains. To demonstrate that one of our new protocols is not only asymptotically optimal but also practical, we perform several probabilistic simulations to estimate the concrete complexity for given parameters. Our simulations show that our protocol significantly improves the per-party communication complexity over the state-of-the-art for practical parameters. Hence, for given bandwidth constraints, our results allow to, e.g., increase the block size, improving the overall throughput of a blockchain.

Many decentralized systems rely on flooding protocols for message dissemination. In such a protocol, the sender of a message sends it to a randomly selected set of peers. These peers again send the message to their randomly selected peers, until every network participant has received the message. This type of protocols clearly fail in face of an adaptive adversary who can simply corrupt all peers of the sender and thereby prevent the message from being delivered. Nevertheless, flooding protocols are commonly used within protocols that aim to be cryptographically secure, most notably in blockchain protocols. While it is possible to revert to static corruptions, this gives unsatisfactory security guarantees, especially in the setting of a blockchain that is supposed to run for an extended period of time. To be able to provide meaningful security guarantees in such settings, we give precise semantics to what we call $\delta$-delayed adversaries in the Universal Composability (UC) framework. Such adversaries can adaptively corrupt parties, but there is a delay of time $\delta$ from when an adversary decides to corrupt a party until they succeed in overtaking control of the party. Within this model, we formally prove the intuitive result that flooding protocols are secure against $\delta$-delayed adversaries when $\delta$ is at least the time it takes to send a message from one peer to another plus the time it takes the recipient to resend the message. To this end, we show how to reduce the adaptive setting with a $\delta$-delayed adversary to a static experiment with an Erdős–Rényi graph. Using the established theory of Erdős–Rényi graphs, we provide upper bounds on the propagation time of the flooding functionality for different neighborhood sizes of the gossip network. More concretely, we show the following for security parameter $\kappa$, point-to-point channels with delay at most $\Delta$, and $n$ parties in total, with a sufficiently delayed adversary that can corrupt any constant fraction of the parties: If all parties send to $\Omega(\kappa)$ parties on average, then we can realize a flooding functionality with maximal delay $\mathcal{O}\bigl(\Delta \cdot \log (n) \bigr)$; and if all parties send to $\Omega\bigl( \sqrt{\kappa n \log (n)} \bigr)$ parties on average, we can realize a flooding functionality with maximal delay $\mathcal{O}(\Delta)$.

In recent years, permisionless blockchains have received a lot of attention both from industry and academia, where substantial effort has been spent to develop consensus protocols that are secure under the assumption that less than half (or a third) of a given resource (e.g., stake or computing power) is controlled by corrupted parties. The security proofs of these consensus protocols usually assume the availability of a network functionality guaranteeing that a block sent by an honest party is received by all honest parties within some bounded time. To obtain an overall protocol that is secure under the same corruption assumption, it is therefore necessary to combine the consensus protocol with a network protocol that achieves this property under that assumption. In practice, however, the underlying network is typically implemented by flooding protocols that are not proven to be secure in the setting where a fraction of the considered total weight can be corrupted. This has led to many so-called eclipse attacks on existing protocols and tailor-made fixes against specific attacks. To close this apparent gap, we present the first practical flooding protocol that provably delivers sent messages to all honest parties after a logarithmic number of steps. We prove security in the setting where all parties are publicly assigned a positive weight and the adversary can corrupt parties accumulating up to a constant fraction of the total weight. This can directly be used in the proof-of-stake setting, but is not limited to it. To prove the security of our protocol, we combine known results about the diameter of Erdős–Rényi graphs with reductions between different types of random graphs. We further show that the efficiency of our protocol is asymptotically optimal. The practicality of our protocol is supported by extensive simulations for different numbers of parties, weight distributions, and corruption strategies. The simulations confirm our theoretical results and show that messages are delivered quickly regardless of the weight distribution, whereas protocols that are oblivious of the parties' weights completely fail if the weights are unevenly distributed. Furthermore, the average message complexity per party of our protocol is within a small constant factor of such a protocol.

We introduce policy-compliant signatures (PCS). A PCS scheme can be used in a setting where a central authority determines a global policy and distributes public and secret keys associated with sets of attributes to the users in the system. If two users, Alice and Bob, have attribute sets that jointly satisfy the global policy, Alice can use her secret key and Bob's public key to sign a message. Unforgeability ensures that a valid signature can only be produced if Alice's secret key is known and if the policy is satisfied. Privacy guarantees that the public keys and produced signatures reveal nothing about the users' attributes beyond whether they satisfy the policy or not. PCS extends the functionality provided by existing primitives such as attribute-based signatures and policy-based signatures, which do not consider a designated receiver and thus cannot include the receiver's attributes in the policies. We describe practical applications of PCS which include controlling transactions in financial systems with strong privacy guarantees (avoiding additional trusted entities that check compliance), as well as being a tool for trust negotiations. We introduce an indistinguishability-based privacy notion for PCS and present a generic and modular scheme based on standard building blocks such as signatures, non-interactive zero-knowledge proofs, and a (predicate-only) predicate encryption scheme. We show that it can be instantiated to obtain an efficient scheme that is provably secure under standard pairing-assumptions for a wide range of policies. We further model PCS in UC by describing the goal of PCS as an enhanced ideal signature functionality which gives rise to a simulation-based privacy notion for PCS. We show that our generic scheme achieves this composable security notion under the additional assumption that the underlying predicate encryption scheme satisfies a stronger, fully adaptive, simulation-based attribute-hiding notion.

In this work, we introduce and construct D-restricted Functional Encryption (FE) for any constant $$D \ge 3$$D≥3, based only on the SXDH assumption over bilinear groups. This generalizes the notion of 3-restricted FE recently introduced and constructed by Ananth et al. (ePrint 2018) in the generic bilinear group model.A $$D=(d+2)$$D=(d+2)-restricted FE scheme is a secret key FE scheme that allows an encryptor to efficiently encrypt a message of the form $$M=(\varvec{x},\varvec{y},\varvec{z})$$M=(x,y,z). Here, $$\varvec{x}\in \mathbb {F}_{\mathbf {p}}^{d\times n}$$x∈Fpd×n and $$\varvec{y},\varvec{z}\in \mathbb {F}_{\mathbf {p}}^n$$y,z∈Fpn. Function keys can be issued for a function $$f=\varSigma _{\varvec{I}= (i_1,..,i_d,j,k)}\ c_{\varvec{I}}\cdot \varvec{x}[1,i_1] \cdots \varvec{x}[d,i_d] \cdot \varvec{y}[j]\cdot \varvec{z}[k]$$f=ΣI=(i1,..,id,j,k)cI·x[1,i1]⋯x[d,id]·y[j]·z[k] where the coefficients $$c_{\varvec{I}}\in \mathbb {F}_{\mathbf {p}}$$cI∈Fp. Knowing the function key and the ciphertext, one can learn $$f(\varvec{x},\varvec{y},\varvec{z})$$f(x,y,z), if this value is bounded in absolute value by some polynomial in the security parameter and n. The security requirement is that the ciphertext hides $$\varvec{y}$$y and $$\varvec{z}$$z, although it is not required to hide $$\varvec{x}$$x. Thus $$\varvec{x}$$x can be seen as a public attribute.D-restricted FE allows for useful evaluation of constant-degree polynomials, while only requiring the SXDH assumption over bilinear groups. As such, it is a powerful tool for leveraging hardness that exists in constant-degree expanding families of polynomials over $$\mathbb {R}$$R. In particular, we build upon the work of Ananth et al. to show how to build indistinguishability obfuscation ($$i\mathcal {O}$$iO) assuming only SXDH over bilinear groups, LWE, and assumptions relating to weak pseudorandom properties of constant-degree expanding polynomials over $$\mathbb {R}$$R.

The existence of secure indistinguishability obfuscators ( $$i\mathcal {O}$$ ) has far-reaching implications, significantly expanding the scope of problems amenable to cryptographic study. All known approaches to constructing $$i\mathcal {O}$$ rely on d-linear maps. While secure bilinear maps are well established in cryptographic literature, the security of candidates for $$d>2$$ is poorly understood.We propose a new approach to constructing $$i\mathcal {O}$$ for general circuits. Unlike all previously known realizations of $$i\mathcal {O}$$ , we avoid the use of d-linear maps of degree $$d \ge 3$$ .At the heart of our approach is the assumption that a new weak pseudorandom object exists. We consider two related variants of these objects, which we call perturbation resilient generator ( $$\varDelta $$ RG) and pseudo flawed-smudging generator ( $$\mathrm {PFG}$$ ), respectively. At a high level, both objects are polynomially expanding functions whose outputs partially hide (or smudge) small noise vectors when added to them. We further require that they are computable by a family of degree-3 polynomials over $$\mathbb {Z}$$ . We show how they can be used to construct functional encryption schemes with weak security guarantees. Finally, we use novel amplification techniques to obtain full security.As a result, we obtain $$i\mathcal {O}$$ for general circuits assuming:Subexponentially secure LWEBilinear Maps $$\mathrm {poly}(\lambda )$$ -secure 3-block-local PRGs $$\varDelta $$ RGs or $$\mathrm {PFG}$$ s