International Association
for Cryptologic Research

We analyze the post-quantum security of succinct interactive arguments constructed from interactive oracle proofs (IOPs) and vector commitment schemes. Specifically, we prove that an interactive variant of the *BCS transformation* is secure in the standard model against quantum adversaries when the vector commitment scheme is collapse binding. Prior work established the post-quantum security of Kilian's succinct interactive argument, a special case of the BCS transformation for one-message IOPs (i.e., PCPs). That analysis is inherently limited to one message because the reduction, like all prior quantum rewinding reductions, aims to extract classical information (a PCP string) from the quantum argument adversary. Our reduction overcomes this limitation by instead extracting a *quantum algorithm* that implements an IOP adversary; representing such an adversary classically may in general require exponential complexity. Along the way we define *collapse position binding*, which we propose as the ``correct'' definition of collapse binding for vector commitment schemes, eliminating shortcomings of prior definitions. As an application of our results, we obtain post-quantum secure succinct arguments, in the standard model (no oracles), with the *best asymptotic complexity known*.

Proofs for machine computation prove the correct execution of arbitrary programs that operate over fixed instruction sets (e.g., RISC-V, EVM, Wasm). A standard approach for proving machine computation is to prove a universal set of constraints that encode the full instruction set at each step of the program execution. This approach incurs a proving cost per execution step on the order of the total sum of instruction constraints for all of the instructions in the set, despite each step of the program only executing a single instruction. Existing proving approaches that avoid this universal cost per step (and incur only the cost of a single instruction's constraints per step) either fail to provide zero-knowledge or rely on recursive proof composition for which security relies on the heuristic instantiation of the random oracle. We present new protocols for proving machine execution that resolve these limitations, enabling prover efficiency on the order of only the executed instructions while achieving zero-knowledge and avoiding recursive proofs. Our core technical contribution is a new primitive that we call a succinct vector lookup argument which enables a prover to build up a machine execution ``on-the-fly''. We propose succinct vector lookups for both univariate polynomial and multivariate polynomial commitments in which vectors are encoded on cosets of a multiplicative subgroup and on subcubes of the boolean hypercube, respectively. We instantiate our proofs for machine computation by integrating our vector lookups with existing efficient, succinct non-interactive proof systems for NP.