Pairing-Based SNARGs with Two Group Elements.

SNARGs are cryptographic primitives that allow a prover to demonstrate membership in an NP language while sending a proof that is much smaller than the witness. In this work, we focus on the succinctness of publicly-verifiable group-based SNARGs, analyzed in a model that combines both a generic (asymmetric) bilinear group (𝔾_1×𝔾_2→𝔾_T) and a random oracle (the GGM + ROM). We construct the first publicly-verifiable SNARG in the GGM + ROM where the proof consists of exactly 2 elements of 𝔾_1 and no additional bits, achieving the smallest proof size among all known publicly verifiable group-based SNARGs. Our security analysis is tight, ensuring that the construction incurs no hidden security losses. Concretely, when instantiated with the BLS12-381 curve for 128-bit security, our scheme yields a proof size of 768 bits, nearly a 2× improvement over the best known pairing-based SNARG. While our scheme is not yet concretely efficient, it demonstrates the feasibility of ultra-short proofs and opens the door to future practical instantiations. Complementing this construction, we establish a new lower bound for group-based SNARGs. We prove that under mild and natural restrictions on the verifier (which are satisfied by all known schemes) no SNARG exists in the Maurer GGM + ROM with a proof that consists of a single group element (assuming one-way functions). This substantially strengthens the lower bound of Groth, which was more restrictive and did not extend to settings with a random oracle.