PoReps: Proofs of Space on Useful Data

A proof-of-replication (PoRep) is an interactive proof system in which a prover defends a publicly verifiable claim that it is dedicating unique resources to storing one or more retrievable replicas of a data file. In this sense a PoRep is both a proof of space (PoS) and a proof of retrievability (PoR). This paper is a foundational study of PoReps, exploring both their capabilities and their limitations. While PoReps may unconditionally demonstrate possession of data, they fundamentally cannot guarantee that the data is stored redundantly. Furthermore, as PoReps are proofs of space, they must rely either on rational time/space tradeoffs or timing bounds on the online prover\u27s runtime. We introduce a rational security notion for PoReps called epsilon-rational replication based on the notion of an epsilon-Nash equilibrium, which captures the property that a server does not gain any significant advantage by storing its data in any other (non-redundant) format. We apply our definitions to formally analyze two recently proposed PoRep constructions based on verifiable delay functions and depth robust graphs. Lastly, we reflect on a notable application of PoReps---its unique suitability as a Nakamoto consensus mechanism that replaces proof-of-work with PoReps on real data, simultaneously incentivizing and subsidizing the cost of file storage

Derecho: Privacy Pools with Proof-Carrying Disclosures

A privacy pool enables clients to deposit units of a cryptocurrency into a shared pool where ownership of deposited currency is tracked via a system of cryptographically hidden records. Clients may later withdraw from the pool without linkage to previous deposits. Some privacy pools also support hidden transfer of currency ownership within the pool. In August 2022, the U.S. Department of Treasury sanctioned Tornado Cash, the largest Ethereum privacy pool, on the premise that it enables illicit actors to hide the origin of funds, citing its usage by the DPRK-sponsored Lazarus Group to launder over \$455 million dollars worth of stolen cryptocurrency. This ruling effectively made it illegal for U.S. persons/institutions to use or accept funds that went through Tornado Cash, sparking a global debate among privacy rights activists and lawmakers. Against this backdrop, we present Derecho, a system that institutions could use to request cryptographic attestations of fund origins rather than naively rejecting all funds coming from privacy pools. Derecho is a novel application of proof-carrying data, which allows users to propagate allowlist membership proofs through a privacy pool\u27s transaction graph. Derecho is backwards-compatible with existing Ethereum privacy pool designs, adds no overhead in gas costs, and costs users only a few seconds to produce attestations

Weak Compression and (In)security of Rational Proofs of Storage

We point out an implicit unproven assumption underlying the security of rational proofs of storage that is related to a concept we call weak randomized compression. This is a class of interactive proofs designed in a security model with a rational prover who is encouraged to store data (possibly in a particular format), as otherwise it either fails verification or does not save any storage costs by deviating (in some cases it may even increase costs by ``wasting the space). Weak randomized compression is a scheme that takes a random seed $r$ and a compressible string $s$ and outputs a compression of the concatenation $r \circ s$. Strong compression would compress $s$ by itself (and store the random seed separately). In the context of a storage protocol, it is plausible that the adversary knows a weak compression that uses its incompressible storage advice as a seed to help compress other useful data it is storing, and yet it does not know a strong compression that would perform just as well. It therefore may be incentivized to deviate from the protocol in order to save space. This would be particularly problematic for proofs of replication, designed to encourage provers to store data in a redundant format, which weak compression would likely destroy. We thus motivate the question of whether weak compression can always be used to efficiently construct strong compression, and find (negatively) that a black-box reduction would imply a universal compression scheme in the random oracle model for all compressible efficiently sampleable sources. Implausibility of universal compression aside, we conclude that constructing this black-box reduction for a class of sources is at least as hard as directly constructing a universal compression scheme for that class

Batching Techniques for Accumulators with Applications to IOPs and Stateless Blockchains

We present batching techniques for cryptographic accumulators and vector commitments in groups of unknown order. Our techniques are tailored for distributed settings where no trusted accumulator manager exists and updates to the accumulator are processed in batches. We develop techniques for non-interactively aggregating membership proofs that can be verified with a constant number of group operations. We also provide a constant sized batch non-membership proof for a large number of elements. These proofs can be used to build the first positional vector commitment (VC) with constant sized openings and constant sized public parameters. As a core building block for our batching techniques we develop several succinct proof systems in groups of unknown order. These extend a recent construction of a succinct proof of correct exponentiation, and include a succinct proof of knowledge of an integer discrete logarithm between two group elements. We use these new constructions to design a stateless blockchain, where nodes only need a constant amount of storage in order to participate in consensus. Further, we show how to use these techniques to reduce the size of IOP instantiations, such as STARKs

Transparent SNARKs from DARK Compilers

We construct a new polynomial commitment scheme for univariate and multivariate polynomials over finite fields, with logarithmic size evaluation proofs and verification time, measured in the number of coefficients of the polynomial. The underlying technique is a Diophantine Argument of Knowledge (DARK), leveraging integer representations of polynomials and groups of unknown order. Security is shown from the strong RSA and the adaptive root assumptions. Moreover, the scheme does not require a trusted setup if instantiated with class groups. We apply this new cryptographic compiler to a restricted class of algebraic linear IOPs, which we call Polynomial IOPs, to obtain doubly-efficient public-coin interactive arguments of knowledge for any NP relation with succinct communication. With linear preprocessing, the online verifier\u27s work is logarithmic in the circuit complexity of the relation. There are many existing examples of Polynomial IOPs (PIOPs) dating back to the first PCP (BFLS, STOC\u2791). We present a generic compilation of any PIOP using our DARK polynomial commitment scheme. In particular, compiling the PIOP from PLONK (GWC, ePrint\u2719), an improvement on Sonic (MBKM, CCS\u2719), yields a public-coin interactive argument with quasi-linear preprocessing, quasi-linear (online) prover time, logarithmic communication, and logarithmic (online) verification time in the circuit size. Applying Fiat-Shamir results in a SNARK, which we call *Supersonic*. Supersonic is also concretely efficient with 10KB proofs and under 100ms verification time for circuits with 1 million gates (estimated for 120-bit security). Most importantly, this SNARK is transparent: it does not require a trusted setup. We obtain zk-SNARKs by applying a hiding variant of our polynomial commitment scheme with zero-knowledge evaluations. Supersonic is the first complete zk-SNARK system that has both a practical prover time as well as asymptotically logarithmic proof size and verification time. The original proof had a significant gap that was discovered by Block et al. (CRYPTO 2021). The new security proof closes the gap and shows that the original protocol with a slightly adjusted parameter is still secure. Towards this goal, we introduce the notion of almost-special-sound protocols which likely has broader applications

BaseFold: Efficient Field-Agnostic Polynomial Commitment Schemes from Foldable Codes

Interactive Oracle Proof of Proximity (IOPPs) are a powerful tool for constructing succinct non-interactive arguments of knowledge (SNARKs) in the random oracle model, which are fast and plausibly post-quantum secure. The Fast Reed Solomon IOPP (FRI) is the most widely used in practice, while tensor-code IOPPs (such as Brakedown) achieve significantly faster prover times at the cost of much larger proofs. IOPPs are used to construct polynomial commitment schemes (PCS), which are not only an important building block for SNARKs but also have a wide range of independent applications. This work introduces Basefold, a generalization of the FRI IOPP to a broad class of linear codes beyond Reed-Solomon, which we call $\textit{foldable linear codes}$. We construct a new family of foldable linear codes, which are a special type of randomly punctured Reed-Muller code, and prove tight bounds on their minimum distance. Finally, we introduce a new construction of a multilinear PCS from any foldable linear code, which is based on interleaving Basefold with the classical sumcheck protocol for multilinear polynomial evaluation. As a special case, this gives a new multilinear PCS from FRI. In addition to these theoretical contributions, the Basefold PCS instantiated with our new foldable linear codes offers a more reasonable tradeoff between prover time, proof size, and verifier time than prior constructions. For instance, for polynomials over a $64$-bit field with $12$ variables, the Basefold prover is faster than both Brakedown and FRI-PCS ($2$ times faster than Brakedown and $3$ times faster than FRI-PCS), and its proof is $4$ times smaller than Brakedown\u27s. On the other hand, for polynomials with $25$ variables, Basefold\u27s prover is $6.5$ times faster than FRI-PCS, it\u27s proof is $2.5$ times smaller than Brakedown\u27s and its verifier is $7.5$ times faster. Using Basefold to compile the Hyperplonk PIOP [CBBZ23] results in an extremely fast implementation of Hyperplonk, which in addition to having competitive performance on general circuits, is particularly fast for circuits with high-degree custom gates (e.g., signature verification and table lookups). Hyperplonk with Basefold is approximately equivalent to the speed of Hyperplonk with Brakedown, but with a proof size that is more than $5$ times smaller. Finally, Basefold maintains performance across a wider variety of field choices than FRI, which requires FFT-friendly fields. Thus, Basefold can have an extremely fast prover compared to SNARKs from FRI for special applications. Benchmarking a circom ECDSA verification circuit with curve secp256k1, Hyperplonk with Basefold has a prover time that is more than $200\times$ faster than with FRI and its proof size is $5.8$ times smaller than Hyperplonk with Brakedown

Efficient polynomial commitment schemes for multiple points and polynomials

We present an enhanced version of the Kate, Zaverucha and Goldberg polynomial commitment scheme [KZG, ASIACRYPT 2010] where a single group element can be an opening proof for multiple polynomials each evaluated at a different arbitrary subset of points. As a sample application we ``plug in\u27\u27 this scheme into the PLONK proving system[GWC, 2019] to obtain improved proof size and prover run time at the expense of additional verifier ${\mathbb{G}}_2$ operations and pairings, and additional ${\mathbb{G}}_2$ SRS elements. We also present a second scheme where the proof consists of two group elements and the verifier complexity is better than previously known batched verification methods for [KZG]

PIEs: Public Incompressible Encodings for Decentralized Storage

We present a new primitive supporting file replication in distributed storage networks (DSNs) called a Public Incompressible Encoding (PIE). PIEs operate in the challenging public DSN setting where files must be encoded and decoded with public randomness—i.e., without encryption—and retention of redundant data must be publicly verifiable. They prevent undetectable data compression, allowing DSNs to use monetary rewards or penalties in incentivizing economically rational servers to properly replicate data. Their definition also precludes critical, demonstrated attacks involving parallelism via ASICs and other custom hardware. Our PIE construction is the first to achieve experimentally validated near-optimal performance—within a factor of 4 of optimal by one metric. It also allows decoding orders of magnitude faster than encoding, unlike other comparable constructions. We achieve this high security and performance using a graph construction called a Dagwood Sandwich Graph (DSaG), built from a novel interleaving of depth-robust graphs and superconcentrators. PIEs\u27 performance makes them appealing for DSNs, such as the proposed Filecoin system and Ethereum data sharding. Conversely, their near-optimality establishes concerning bounds on the practical financial and energy costs of DSNs allowing arbitrary data

Iron: Functional Encryption using Intel SGX

Functional encryption (FE) is an extremely powerful cryptographic mechanism that lets an authorized entity compute on encrypted data, and learn the results in the clear. However, all current cryptographic instantiations for general FE are too impractical to be implemented. We build Iron, a practical and usable FE system using Intel\u27s recent Software Guard Extensions (SGX). We show that Iron can be applied to complex functionalities, and even for simple functions, outperforms the best known cryptographic schemes. We argue security by modeling FE in the context of hardware elements, and prove that Iron satisfies the security model

Deterministic Soluble Model of Coarsening

We investigate a 3-phase deterministic one-dimensional phase ordering model in which interfaces move ballistically and annihilate upon colliding. We determine analytically the autocorrelation function A(t). This is done by computing generalized first-passage type probabilities P_n(t) which measure the fraction of space crossed by exactly n interfaces during the time interval (0,t), and then expressing the autocorrelation function via P_n's. We further reveal the spatial structure of the system by analyzing the domain size distribution.Comment: 5 pages, RevTeX fil