Paper 2024/1434 Original: eprint.iacr.org/2024/1434 ↗

Untangling the Security of Kilian's Protocol: Upper and Lower Bounds

Abstract

Sigma protocols are elegant cryptographic proofs that have become a cornerstone of modern cryptography. A notable example is Schnorr's protocol, a zero-knowledge proof-of-knowledge of a discrete logarithm. Despite extensive research, the security of Schnorr's protocol in the standard model is not fully understood. In this paper we study Kilian's protocol, an influential public-coin interactive protocol that, while not a sigma protocol, shares striking similarities with sigma protocols. The first example of a succinct argument, Kilian's protocol is proved secure via rewinding, the same idea used to prove sigma protocols secure. In this paper we show how, similar to Schnorr's protocol, a precise understanding of the security of Kilian's protocol remains elusive. We contribute new insights via upper bounds and lower bounds. - Upper bounds. We establish the tightest known bounds on the security of Kilian's protocol in the standard model, via strict-time reductions and via expected-time reductions. Prior analyses are strict-time reductions that incur large overheads or assume restrictive properties of the PCP underlying Kilian's protocol. - Lower bounds. We prove that significantly improving on the bounds that we establish for Kilian's protocol would imply improving the security analysis of Schnorr's protocol beyond the current state-of-the-art (an open problem). This partly explains the difficulties in obtaining tight bounds for Kilian's protocol.