Succinct Non-subsequence Arguments

Lookup arguments have recently attracted a lot of developments due to their applications in the constructions of succinct non-interactive arguments of knowledge (SNARKs). A closely related topic is subsequence arguments in which one can prove that string \(\textbf{s}\) is a subsequence of another string \(\textbf{t}\) , i.e., deleting some characters in \(\textbf{t}\) can achieve \(\textbf{s}\) . A dual notion, namely, non-subsequence arguments, is to prove that \(\textbf{s}\) is not a subsequence of \(\textbf{t}\) . These problems have a lot of important applications in DNA sequence analysis, internet of things, blockchains, natural language processing, speech recognition, etc. However, despite their applications, they are not well-studied in cryptography, especially succinct arguments for non-subsequences with efficient proving time and sublinear verification time. In this work, we propose the first succinct non-subsequence argument. Our solution applies the sumcheck protocol and is instantiable by any multivariate polynomial commitment schemes (PCSs). We achieve an efficient prover whose running time is linear in the size of sequences \(\textbf{s}\) , \(\textbf{t}\) and their respective alphabet \(\varSigma \) . Our proof is succinct and the verifier time is sublinear assuming the employed PCS has succinct commitments and sublinear verification time. When instantiating with Sona PCS (EUROCRYPT’24), we achieve proof size \(\mathcal {O}(\log _2|\textbf{s}| + \log _2|\textbf{t}|+\log _2|\varSigma |)\) , prover time \(\mathcal {O}(|\textbf{s}|+|\textbf{t}|+|\varSigma |)\) and verifier time \(\mathcal {O}(\sqrt{|\textbf{s}|}+\sqrt{|\textbf{t}|}+\sqrt{|\varSigma |})\) . Extending our technique, we can achieve a batch subsequence argument for proving in batch k interleaving subsequence and non-subsequence arguments without proof size suffering a linear blow-up in k.