Bulletproofs*: Verifier-Efficient Arithmetic Circuit Proofs via Folding

Emanuele Scala; Daniele Bartoli

Paper 2026/586

Bulletproofs*: Verifier-Efficient Arithmetic Circuit Proofs via Folding

Emanuele Scala

, Independent Researcher

Daniele Bartoli

, University of Perugia

Abstract

We present Bulletproofs* (BP*, BulletproofsStar), a folding scheme for arithmetic circuit proofs under standard assumptions and without preprocessing, i.e., for the arithmetic circuit satisfiability language of Bulletproofs (S&P 18), following the recipe of ProtoStar (ePrint 2023/620). To this end, we first adapt the algebraic verifiers of the arithmetic circuit proof of Bulletproofs to the algebraic form required by ProtoStar, and prove that the modified protocol remains secure. Then, we design the Bulletproofs* folding scheme that is complete and knowledge-sound. Finally, we analyze the resulting verifier cost after the folding-to-IVC transformation. The result shows an asymptotic linear gain compared to repeated invocations of the monolithic Bulletproofs verifier.

Note: Preprint. Revised version. This revision adds the complete proofs of Theorem 1 and Theorem 2, introduces the Commit-and-Open variant, and updates the folding construction, complexity analysis, and comparison accordingly.

Metadata

Available format(s): PDF
Category: Cryptographic protocols
Publication info: Preprint.
Keywords: Folding Scheme Bulletproofs ProtoStar Zero-Knowledge Proofs Incremental Verifiable Computation Recursive Proofs
Contact author(s): emsczkp @ proton me
daniele bartoli @ unipg it
History: 2026-06-24: revised; 2026-03-24: received; See all versions
Short URL: https://ia.cr/2026/586
License: CC BY

BibTeX

@misc{cryptoeprint:2026/586,
      author = {Emanuele Scala and Daniele Bartoli},
      title = {Bulletproofs*: Verifier-Efficient Arithmetic Circuit Proofs via Folding},
      howpublished = {Cryptology {ePrint} Archive, Paper 2026/586},
      year = {2026},
      url = {https://eprint.iacr.org/2026/586}
}