International Association
for Cryptologic Research

This paper introduces Lasso, a new family of lookup arguments, which allow an untrusted prover to commit to a vector $a \in \mathbb{F}^m$ and prove that all entries of $a$ reside in some predetermined table $t \in \mathbb{F}^n$. Lasso's performance characteristics unlock the so-called ``lookup singularity''. Lasso works with any multilinear polynomial commitment scheme, and provides the following efficiency properties. (1) For $m$ lookups into a table of size $n$, Lasso's prover commits to just $m+n$ field elements. Moreover, the committed field elements are \emph{small}, meaning that, no matter how big the field $\mathbb{F}$ is, they are all in the set $\{0, \dots, m\}$. When using a multiexponentiation-based commitment scheme, this results in the prover's costs dominated by only $O(m+n)$ group \emph{operations} (e.g., elliptic curve point additions), plus the cost to prove an evaluation of a multilinear polynomial whose evaluations over the Boolean hypercube are the table entries. This represents a significant improvement in prover costs over prior lookup arguments (e.g., plookup, Halo2's lookups, lookup arguments based on logarithmic derivatives). (2) Unlike all prior lookup arguments, if the table $t$ is structured (in a precise sense that we define), then no party needs to commit to $t$, enabling the use of much larger tables than prior works (e.g., of size $2^{128}$ or larger). Moreover, Lasso's prover only ``pays'' in runtime for table entries that are accessed by the lookup operations. This applies to tables commonly used to implement range checks, bitwise operations, big-number arithmetic, and even transitions of a full-fledged CPU such as RISC-V. Specifically, for any integer parameter $c>1$, Lasso's prover's dominant cost is committing to $3 \cdot c \cdot m + c \cdot n^{1/c}$ field elements. Furthermore, all these field elements are ``small'', meaning they are in the set $\{0, \dots, \max\{m, n^{1/c}, q\}-1\}$, where $q$ is the maximum value in any of the sub-tables that collectively capture $t$ (in a precise manner that we define).

Succinct Non-interactive Arguments of Knowledge (SNARKs) allow an untrusted prover to establish that it correctly ran some “witness-checking procedure” on a witness. A zkVM (short for zero-knowledge Virtual Machine) is a SNARK that allows the witness-checking procedure to be specified as a computer program written in the assembly language of a specific instruction set architecture (ISA). A front-end converts computer programs into a lower-level representation such as an arithmetic circuit or generalization thereof. A SNARK for circuit- satisfiability can then be applied to the resulting circuit. We describe a new front-end technique called Jolt that applies to a variety of ISAs. Jolt arguably realizes a vision called the lookup singularity, which seeks to produce circuits that only perform lookups into pre-determined lookup tables. The circuits output by Jolt primarily perform lookups into a gigantic lookup table, of size more than 2^128, which depends only on the ISA. The validity of the lookups are proved via a new lookup argument described in a companion work called Lasso. Although size 2^128 tables are vastly too large to materialize in full, the tables arising in Jolt are structured, avoiding costs that grow linearly with the table size. We describe performance and auditability benefits of Jolt compared to prior zkVMs, focusing on the popular RISC-V ISA as a concrete example. The dominant cost for the Jolt prover applied to this ISA (on 64-bit data types) is cryptographically committing to about six 256-bit field elements per step of the RISC-V CPU. This compares favorably to prior zkVM provers, even those focused on far simpler VMs.

We introduce HyperNova, a new recursive argument for proving incremental computations whose steps are expressed with CCS (Setty et al. ePrint 2023/552), a customizable constraint system that simultaneously generalizes Plonkish, R1CS, and AIR without overheads. HyperNova makes four contributions, each resolving a major problem in the area of recursive arguments. First, it provides a folding scheme for CCS where the prover’s cryptographic cost is a single multi-scalar multiplication (MSM) of size equal to the number of variables in the constraint system, which is optimal when using an MSM-based commitment scheme. The folding scheme can fold multiple instances at once, making it easier to build generalizations of IVC such as PCD. Second, when proving program executions on stateful machines (e.g., EVM, RISC-V), the cost of proving a step of a program is proportional only to the size of the circuit representing the instruction invoked by the program step (“a la carte” cost profile). Third, we show how to achieve zero-knowledge for “free” and without the need to employ zero-knowledge SNARKs. Fourth, we show how to efficiently instantiate HyperNova over a cycle of elliptic curves. For this, we provide a general technique, which we refer to as CycleFold, that applies to all modern folding-scheme-based recursive arguments.

This paper introduces a SNARK called Brakedown. Brakedown targets R1CS, a popular NP-complete problem that generalizes circuit-satisfiability. It is the first built system that provides a linear-time prover, meaning the prover incurs O(N) finite field operations to prove the satisfiability of an N-sized R1CS instance. Brakedown’s prover is faster, both concretely and asymptotically, than prior SNARK implementations. It does not require a trusted setup and may be post-quantum secure. Furthermore, it is compatible with arbitrary finite fields of sufficient size; this property is new among built proof systems with sublinear proof sizes. To design Brakedown, we observe that recent work of Bootle, Chiesa, and Groth (BCG, TCC 2020) provides a polynomial commitment scheme that, when combined with the linear-time interactive proof system of Spartan (CRYPTO 2020), yields linear-time IOPs and SNARKs for R1CS (a similar theoretical result was previously established by BCG, but our approach is conceptually simpler, and crucial for achieving high-speed SNARKs). A core ingredient in the polynomial commitment scheme that we distill from BCG is a linear-time encodable code. Existing constructions of such codes are believed to be impractical. Nonetheless, we design and engineer a new one that is practical in our context. We also implement a variant of Brakedown that uses Reed-Solomon codes instead of our linear-time encodable codes; we refer to this variant as Shockwave. Shockwave is not a linear-time SNARK, but it provides shorter proofs and lower verification times than Brakedown, and also provides a faster prover than prior plausibly post-quantum SNARKs.

We introduce a new approach to realize incrementally verifiable computation (IVC), in which the prover recursively proves the correct execution of incremental computations of the form y=F^{(\ell)}(x), where F is a (potentially non-deterministic) computation, x is the input, y is the output, and \ell > 0. Unlike prior approaches to realize IVC, our approach avoids succinct non-interactive arguments of knowledge (SNARKs) entirely (and arguments of knowledge in general). Instead, we introduce and employ folding schemes, a weaker, simpler, and more efficiently-realizable primitive, which reduces the task of checking two instances in some relation to the task of checking a single instance. We construct a folding scheme for NP and show that it implies an IVC scheme with improved efficiency characteristics: (1) the "recursion overhead" (i.e., the number of steps that the prover proves in addition to proving the execution of F) is a constant and it is dominated by two group scalar multiplications expressed as a circuit (this is the smallest recursion overhead in the literature) and (2) the prover's work at each step is dominated by two multiexponentiations of size O(|F|), providing the fastest prover in the literature. The size of a proof is O(|F|) group elements, but we show that using a variant of an existing zkSNARK, the prover can prove the knowledge of a valid proof succinctly and in zero-knowledge with O(\log{|F|}) group elements. Finally, our approach neither requires a trusted setup nor FFTs, so it can be instantiated efficiently with any cycles of elliptic curves where DLOG is hard.

This paper introduces Spartan, a new family of zero-knowledge succinct non- interactive arguments of knowledge (zkSNARKs) for the rank-1 constraint satisfiabil- ity (R1CS), an NP-complete language that generalizes arithmetic circuit satisfiability. A distinctive feature of Spartan is that it offers the first zkSNARKs without trusted setup (i.e., transparent zkSNARKs) for NP where verifying a proof incurs sub-linear costs—without requiring uniformity in the NP statement’s structure. Furthermore, Spartan offers zkSNARKs with a time-optimal prover, a property that has remained elusive for nearly all zkSNARKs in the literature. To achieve these results, we introduce new techniques that we compose with the sum-check protocol, a seminal interactive proof protocol: (1) computation commit- ments, a primitive to create a succinct commitment to a description of a computation; this technique is crucial for a verifier to achieve sub-linear costs after investing a one-time, public computation to preprocess a given NP statement; (2) SPARK, a cryptographic compiler to transform any existing extractable polynomial commitment scheme for multilinear polynomials to one that efficiently handles sparse multilinear polynomials; this technique is critical for achieving a time-optimal prover; and (3) a compact encoding of an R1CS instance as a low-degree polynomial. The end result is a public-coin succinct interactive argument of knowledge for NP (which can be viewed as a succinct variant of the sum-check protocol); we transform it into a zkSNARK using prior techniques. By applying SPARK to different commitment schemes, we obtain several zkSNARKs where the verifier’s costs and the proof size range from $O(log^2{n})$ to $O(\sqrt{n})$ depending on the underlying commitment scheme ($n$ denotes the size of the NP statement). These schemes do not require a trusted setup except for one that requires a universal trusted setup. We implement Spartan as a library in about 8,000 lines of Rust. We use the library to build a transparent zkSNARK in the random oracle model where security holds under the discrete logarithm assumption. We experimentally evaluate it and compare with recent zkSNARKs for R1CS instance sizes up to 220 constraints. Among schemes without trusted setup, Spartan offers the fastest prover with speedups of 36--$152\times$ depending on the baseline, produces proofs that are shorter by 1.2--$416\times$, and incurs the lowest verification times with speedups of 3.6--$1326\times$. When compared to the state-of-the-art zkSNARK with trusted setup, Spartan’s prover is $2\times$ faster for arbitrary R1CS instances and $16\times$ faster for data-parallel workloads.