Cougar: Cubic Root Verifier Inner Product Argument under Discrete Logarithm Assumption

Hyeonbum Lee, Seunghun Paik, Hyunjung Son, Jae Hong Seo

Abstract: An inner product argument (IPA) is a cryptographic proof system that serves as a fundamental building block for various applications, such as zero knowledge proofs and verifiable computation. Bulletproofs (IEEE S&P 2018), a well-known IPA under the discrete logarithm (DL) assumption, features a short, logarithmically-sized proof, making it suitable for blockchain applications. However, its major drawback is the linear verifier cost (O(N)), which presents a significant bottleneck in settings like verifiable computation. To address this, recent advancements have successfully reduced the verification complexity to square-root order (O(√N)) under the same assumption (e.g., Asiacrypt 2022, IEEE TIFS). In thiswork, we propose Cougar, a novel IPAthat breaks this square-root barrier to achieve an unprecedented cubic-root verifier complexity (O(3√N)), while strictly maintaining the compact logarithmic proof size (O(log N)) characteristic of Bulletproofs. To achieve this, Cougar introduces a generalized two-tier commitment framework combined with a disjoint interpolation strategy for efficient consistency checks. We implemented Cougar in Rust and performed a comprehensive benchmarking against Bulletproofs and Leopard (IEEE TIFS). Our evaluation demonstrates that while Cougar incurs a moderate increase in prover overhead, its verification time scales significantly better for large instances. Concretely, for a witness size of N = 220, Cougar achieves a 50× verification speed-up over Bulletproofs and exhibits a superior asymptotic growth rate compared to existing sublinear IPAs.

External IDs:doi:10.1109/access.2026.3654539