Abstract CairoZero is a programming language for running decentralized applications (dApps) at scale. Programs written in the CairoZero language are compiled to machine code for the Cairo CPU architecture and cryptographic protocols are used to verify the results of execution efficiently on blockchain. We explain how we have extended the CairoZero compiler with tooling that enables users to prove, in the Lean 3 proof assistant, that compiled code satisfies high-level functional specifications. We demonstrate the success of our approach by verifying primitives for computation with the secp256k1 and secp256r1 curves over a large finite field as well as the validation of cryptographic signatures using the former. We also verify a mechanism for simulating a read-write dictionary data structure in a read-only setting. Finally, we reflect on our methodology and discuss some of the benefits of our approach.

Cryptographic interactive proof systems provide an efficient and scalable means of verifying the results of computation on blockchain. A prover constructs a proof, off-chain, that the execution of a program on a given input terminates with a certain result. The prover then publishes a certificate that can be verified efficiently and reliably modulo commonly accepted cryptographic assumptions. The method relies on an algebraic encoding of execution traces of programs. Here we report on a verification of the correctness of such an encoding of the Cairo model of computation with respect to the STARK interactive proof system, using the Lean 3 proof assistant.

Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worst-case-to-average-case reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. The previous state of the art by [Ben-Sasson, Kopparty, Saraf, CCC 2018] showed that if some member of an affine space U is δ-far in relative Hamming distance from a linear code V - this is the worst-case assumption - then most elements of U are almost-δ-far from V - this is the average case. However, this result was known to hold only below the "double Johnson" function of the relative distance δ_V of the code V, i.e., only when δ < 1-(1-δ_V)^(1/4). First, we increase the soundness-bound to the "one-and-a-half Johnson" function of δ_V and show that the average distance of U from V is nearly δ for any worst-case distance δ smaller than 1-(1-δ_V)^(1/3). This bound is tight, which is somewhat surprising because the one-and-a-half Johnson function is unfamiliar in the literature on error correcting codes. To improve soundness further for Reed Solomon codes we sample outside the box. We suggest a new protocol in which the verifier samples a single point z outside the box D on which codewords are evaluated, and asks the prover for the value at z of the interpolating polynomial of a random element of U. Intuitively, the answer provided by the prover "forces" it to choose one codeword from a list of "pretenders" that are close to U. We call this technique Domain Extending for Eliminating Pretenders (DEEP). The DEEP method improves the soundness of the worst-case-to-average-case reduction for RS codes up their list decoding radius. This radius is bounded from below by the Johnson bound, implying average distance is approximately δ for all δ < 1-(1-δ_V)^(1/2). Under a plausible conjecture about the list decoding radius of Reed-Solomon codes, average distance from V is approximately δ for all δ. The DEEP technique can be generalized to all linear codes, giving improved reductions for capacity-achieving list-decodable codes. Finally, we use the DEEP technique to devise two new protocols: - An Interactive Oracle Proof of Proximity (IOPP) for RS codes, called DEEP-FRI. The soundness of the protocol improves upon that of the FRI protocol of [Ben-Sasson et al., ICALP 2018] while retaining linear arithmetic proving complexity and logarithmic verifier arithmetic complexity. - An Interactive Oracle Proof (IOP) for the Algebraic Linking IOP (ALI) protocol used to construct zero knowledge scalable transparent arguments of knowledge (ZK-STARKs) in [Ben-Sasson et al., eprint 2018]. The new protocol, called DEEP-ALI, improves soundness of this crucial step from a small constant < 1/8 to a constant arbitrarily close to 1.