Paper 2026/2165 Original: eprint.iacr.org/2026/2165 ↗

Argo: Leaking a Secret from an Invalid Groth16 Proof

Abstract

We present Argo, a randomized encoding scheme for the invalidity of pairing-based SNARKs like Groth16. Argo allows an encoder to encode a proof such that a decoder can derive a secret if the proof is invalid. We prove the selective security of invalidity Argo, as well as the adaptive security of validity Argo, under DDH in one source group of a bilinear pairing in the standard model, a standard assumption that holds in the generic group model used to analyze Groth16. We show how to couple invalidity Argo with a projective garbling scheme like Argo MAC or Duty-Free Bits to construct a garbled circuit for Groth16 invalidity as a function of the proof bits, and how to use it with BitVM to construct a permissionless, optimistic Groth16 verifier for Bitcoin: a decoder proves a proof invalid by publishing a secret, and the decoder can be anyone, not necessarily known in advance. When instantiated with BN254, Argo has an offline encoding size of about 21 kB, a thousand times smaller than the projective garbling scheme, and an end-to-end protocol size more than three orders of magnitude smaller than existing Yao-based approaches. Generating an Argo + Argo MAC garbled circuit takes tens of milliseconds on a laptop, compared to minutes for a Yao GC. Argo is the first protocol that makes permissionless, optimistic SNARK verification on Bitcoin practical.