International Association
for Cryptologic Research

<p>A verifiable delay function (VDF) is an important tool used for adding delay in decentralized applications. This paper surveys and compares two beautiful verifiable delay functions, one due to Pietrzak, and the other due to Wesolowski, In addition, we provide a new computational proof of security for one of them, present an attack on an incorrect implementation of the other, and compare the complexity assumptions needed for both schemes. </p>

Verifiable Delay Functions (VDF) are a class of cryptographic primitives aiming to guarantee a minimum computation time, even for an adversary with massive parallel computational power. They are useful in blockchain protocols, and several practical candidates have been proposed based on exponentiation in a large finite field: Sloth++, Veedo, MinRoot. The underlying assumption of these constructions is that computing an exponentiation x^e requires at least log2(e) sequential multiplications. In this work, we analyze the security of these algebraic VDF candidates. In particular, we show that the latency of exponentiation can be reduced using parallel computation, against the preliminary assumptions.

This works introduces BaseFold, a new field-agnostic Polynomial Commitment Scheme (PCS) for multilinear polynomials that has O(log^{2}(n)) verifier costs and O(n log n) prover time. An important application of a multilinear PCS is constructing Succinct Non-interactive Arguments (SNARKs) from multilinear polynomial interactive oracle proofs (PIOPs). Furthermore, field-agnosticism is a major boon to SNARK efficiency in applications that require (or benefit from) a certain field choice. Our inspiration for BaseFold is the Fast Reed-Solomon Interactive-Oracle Proof of Proximity (FRI IOPP), which leverages two properties of Reed-Solomon (RS) codes defined over FFT-Friendly fields: O(n log n) encoding time, and a second property that we call foldability. We first introduce a generalization of the FRI IOPP that works over any foldable linear code in linear time. Second, we construct a new family of linear codes which we call random foldable codes, that are a special type of punctured Reed-Muller codes, and prove tight bounds on their minimum distance. Unlike RS codes, our new codes are foldable and have O(n log n) encoding time over any sufficiently large field. Finally, we construct a new multilinear PCS by carefully interleaving our IOPP with the classical sumcheck protocol, which also gives a new multilinear PCS from FRI. BaseFold is 2-3 times faster than prior multilinear PCS constructions from FRI when defined over the same finite field. More significantly, using Hyperplonk (Eurocrypt, 2022) as a multilinear PIOP backend for apples-to-apples comparison, we show that BaseFold results in a SNARK that has better concrete efficiency across a range of field choices than with any prior multilinear PCS in the literature. Hyperplonk with Basefold has a proof size that is more than 10 times smaller than Hyperplonk with Brakedown and its verifier is over 30 times faster for circuits with more than 2^{20} gates. Compared to FRI, Hyperplonk with Basefold retains efficiency over any sufficiently large field. For illustration, with BaseFold we can prove ECDSA signature verification over the secp256k1 curve more than 20 times faster than Hyperplonk with FRI and the verifier is also twice as fast. Proofs of signature verification have many useful applications, including offloading blockchain transactions and enabling anonymous credentials over the web.

We present Orbweaver, the first plausibly post-quantum functional commitment to achieve quasilinear prover time together with O(log(n)) proof size and O(log(n)loglog(n)) verifier time. Orbweaver enables evaluation of linear maps on committed vectors over cyclotomic rings or the integers. It is extractable, preprocessing, non-interactive, structure-preserving, amenable to recursive composition, and supports logarithmic public proof aggregation. The security of our scheme is based on the k-R-ISIS assumption (and its knowledge counterpart), whereby we require a trusted setup to generate a universal structured reference string. We additionally use Orbweaver to construct a succinct polynomial commitment for integer polynomials.

We show that for $x <-[0,2^\lambda)^u$ and any integer $N$ the probability that $f(x)=0 mod N$ for any non-zero multilinear polynomial $f \in Z[X_1, ...,X_u]$, co-prime to $N$ is inversely proportional to $N$. As a corollary we show that if $log_2 N≥log_2(2 u)\lambda+8u^2 $ then the probability is bounded by $(u+1)/(2^\lambda)$. We also give tighter numerically derived bounds, showing that if $log_2 N≥418 $, and $u ≤ 20$ the probability is bounded by $u/(2^\lambda)+2^{-120}$. We then apply this Multilinear Composite Schwartz-Zippel Lemma (LCSZ) to resolve an open problem in the literature on succinct arguments: that the Bulletproofs protocol for linear relations over classical Pedersen commitments in prime-order groups remains knowledge sound when generalized to commitment schemes that are binding only over short integer vectors. In particular, this means that the Bulletproofs protocol can be instantiated with plausibly post-quantum commitments from lattice hardness assumptions (SIS/R-SIS/M-SIS). It can also be instantiated with commitments based on groups of unknown order (GUOs), in which case the verification time becomes logarithmic instead of linear time. Prior work on lattice-based Bulletproofs (Crypto 2020) and its extensions required modifying the protocol to sample challenges from special sets of polynomial size. This results in a non-negligible knowledge error, necessitating parallel repetition to amplify soundness, which not only impacts efficiency but also poses issues for the Fiat-Shamir transform. Our analysis shows knowledge soundness for the original Bulletproofs protocol with the exponential-size integer challenge set $[0,2^\lambda]$ and thus achieves a negligible soundness error without repetition, circumventing a previous impossibility result (Crypto 2021). Our analysis also closes a critical gap in the original security proof of DARK, a GUO-based polynomial commitment scheme (Eurocrypt 2020). Along the way to achieving our result we also define Almost Special Soundness (AMSS), a generalization of Special-Soundness. Our main result is divided into two parts: (1) that the Bulletproofs protocol over generalized commitments is AMSS, and (2) that AMSS implies knowledge soundness. This framework serves to simplify the application of our analytical techniques to protocols beyond Bulletproofs in the future.

Polynomial commitment schemes (PCS) have recently been in the spotlight for their key role in building SNARKs. A PCS provides the ability to commit to a polynomial over a finite field and prove its evaluation at points. A *succinct* PCS has commitment size and evaluation proof size sublinear in the degree of the polynomial. An *efficient* PCS has sublinear proof verification. Any efficient and succinct PCS can be used to construct a SNARK with similar security and efficiency characteristics (in the random oracle model). Proof-carrying data (PCD) enables a set of parties to carry out an indefinitely long distributed computation where every step along the way is accompanied by a proof of correctness. It generalizes *incrementally verifiable computation* and can even be used to construct SNARKs. Until recently, however, the only known method for constructing PCD required expensive SNARK recursion. A system called *Halo* first demonstrated a new methodology for building PCD without SNARKs, exploiting an aggregation property of the *Bulletproofs* inner-product argument. The construction was *heuristic* because it makes non-black-box use of a concrete instantiation of the Fiat-Shamir transform. We expand upon this methodology to show that PCD can be (heuristically) built from any homomorphic polynomial commitment scheme (PCS), even if the PCS evaluation proofs are neither succinct nor efficient. In fact, the Halo methodology extends to any PCS that has an even more general property, namely the ability to aggregate linear combinations of commitments into a new succinct commitment that can later be opened to this linear combination. Our results thus imply new constructions of SNARKs and PCD that were not previously described in the literature and serve as a blueprint for future constructions as well.

We construct a new polynomial commitment scheme for univariate and multivariate polynomials over finite fields, with public-coin evaluation proofs that have logarithmic communication and verification cost 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 assumption. 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 in order to obtain doubly-efficient public-coin IPs with succinct communication and witness-extended emulation for any NP relation. Allowing for linear preprocessing, the online verifier's work is logarithmic in the circuit complexity of the relation. Concretely, we obtain quasi-linear prover time when compiling the IOP employed in Sonic(MBKM, CCS 19). Applying the Fiat-Shamir transform in the random oracle model results in a SNARK system with quasi-linear preprocessing, quasi-linear (online) prover time, logarithmic proof size, and logarithmic (online) verification time for arbitrary circuits. The SNARK is also concretely efficient with 8.4KB proofs and 75ms verification time for circuits with 1 million gates. Most importantly, this SNARK is transparent: it does not require a trusted setup. We also obtain zk-SNARKs by applying a variant of our polynomial commitment scheme that is hiding and offers zero-knowledge evaluation proofs. This construction is the first transparent zk-SNARK that has both a practical prover time as well as strictly logarithmic proof size and verification time. We call our system Supersonic.

We construct a concretely practical proof-of-space (PoS) with arbitrarily tight security based on stacked depth robust graphs and constant-degree expander graphs. A proof-of-space (PoS) is an interactive proof system where a prover demonstrates that it is persistently using space to store information. A PoS is arbitrarily tight if the honest prover uses exactly N space and for any $$\epsilon > 0$$ϵ>0 the construction can be tuned such that no adversary can pass verification using less than $$(1-\epsilon ) N$$(1-ϵ)N space. Most notably, the degree of the graphs in our construction are independent of $$\epsilon $$ϵ, and the number of layers is only $$O(\log (1/\epsilon ))$$O(log(1/ϵ)). The proof size is $$O(d/\epsilon )$$O(d/ϵ). The degree d depends on the depth robust graphs, which are only required to maintain $$\varOmega (N)$$Ω(N) depth in subgraphs on 80% of the nodes. Our tight PoS is also secure against parallel attacks.Tight proofs of space are necessary for proof-of-replication (PoRep), which is a publicly verifiable proof that the prover is dedicating unique resources to storing one or more retrievable replicas of a specified file. Our main PoS construction can be used as a PoRep, but data extraction is as inefficient as replica generation. We present a second variant of our construction called ZigZag PoRep that has fast/parallelizable data extraction compared to replica generation and maintains the same space tightness while only increasing the number of levels by roughly a factor two.

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 circumvent an impossibility result for Sigma-protocols in these groups by using a short trapdoor-free CRS. We use these new accumulator and vector commitment 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. The full version of the paper is available online [BBF18b].

We study the problem of building a verifiable delay function (VDF). A $$\text {VDF}$$VDFrequires a specified number of sequential steps to evaluate, yet produces a unique output that can be efficiently and publicly verified. $$\text {VDF}$$VDFs have many applications in decentralized systems, including public randomness beacons, leader election in consensus protocols, and proofs of replication. We formalize the requirements for $$\text {VDF}$$VDFs and present new candidate constructions that are the first to achieve an exponential gap between evaluation and verification time.