{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T09:10:06Z","timestamp":1752138606251,"version":"3.41.2"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,7,9]],"date-time":"2025-07-09T00:00:00Z","timestamp":1752019200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,9]],"date-time":"2025-07-09T00:00:00Z","timestamp":1752019200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Beijing Natural Science Foundation","award":["L222050"],"award-info":[{"award-number":["L222050"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62372020","72031001"],"award-info":[{"award-number":["62372020","72031001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Innovation Program of Shanghai Municipal Education Commission","award":["2021-01-07-00-08-E00101"],"award-info":[{"award-number":["2021-01-07-00-08-E00101"]}]},{"name":"\u201cDigital Silk Road\u201d Shanghai International Joint Lab of Trustworthy Intelligent Software","award":["22510750100"],"award-info":[{"award-number":["22510750100"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cybersecurity"],"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Inner product arguments are core building blocks of numerous cryptographic primitives and therefore minimizing their complexity is a central goal in this research area. In this paper, we follow the work of Kim et al. (ASIACRYPT\u201922) and propose the first inner product argument having sublogarithmic communication complexity and sub-square-root verifier complexity simultaneously. We first devise a new subvector combination method for recursion and utilize an aggregated multi-exponentiation argument to prove some committed group elements are valid. We then modify the commitment keys in inner product arguments to be structured and reduce the verifier complexity by delegating the costly computations to the prover. Compared with the state-of-the-art inner product arguments, our protocol is highly competitive in terms of asymptotic complexity.<\/jats:p>","DOI":"10.1186\/s42400-024-00304-x","type":"journal-article","created":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T08:32:20Z","timestamp":1752136340000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Efficient inner product arguments with sublogarithmic proof and sub-square-root verifier"],"prefix":"10.1186","volume":"8","author":[{"given":"Zibo","family":"Zhou","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1358-4933","authenticated-orcid":false,"given":"Zongyang","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jianwei","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Haifeng","family":"Qian","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,7,9]]},"reference":[{"issue":"2","key":"304_CR1","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1007\/s00145-014-9196-7","volume":"29","author":"M Abe","year":"2016","unstructured":"Abe M, Fuchsbauer G, Groth J, Haralambiev K, Ohkubo M (2016) Structure-preserving signatures and commitments to group elements. J Cryptol 29(2):363\u2013421","journal-title":"J Cryptol"},{"key":"304_CR2","doi-asserted-by":"crossref","unstructured":"Attema T, Cramer R (2020) Compressed \u03a3-protocol theory and practical application to plug & play secure algorithmics. In: Micciancio D, Ristenpart T (eds) CRYPTO, vol 12172. LNCS. Springer, Cham, pp 513\u2013543","DOI":"10.1007\/978-3-030-56877-1_18"},{"key":"304_CR3","first-page":"2025","volume-title":"CCS","author":"R Bhadauria","year":"2020","unstructured":"Bhadauria R, Fang Z, Hazay C, Venkitasubramaniam M, Xie T, Zhang Y (2020) Ligero++: a new optimized sublinear IOP. In: Ligatti J, Ou X, Katz J, Vigna G (eds) CCS. ACM, New York, pp 2025\u20132038"},{"issue":"2","key":"304_CR4","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1007\/s00145-007-9005-7","volume":"21","author":"D Boneh","year":"2008","unstructured":"Boneh D, Boyen X (2008) Short signatures without random oracles and the SDH assumption in bilinear groups. J Cryptol 21(2):149\u2013177","journal-title":"J Cryptol"},{"key":"304_CR5","series-title":"LNCS","first-page":"327","volume-title":"EUROCRYPT","author":"J Bootle","year":"2016","unstructured":"Bootle J, Cerulli A, Chaidos P, Groth J, Petit C (2016) Efficient zero-knowledge arguments for arithmetic circuits in the discrete log setting. In: Fischlin M, Coron J (eds) EUROCRYPT, vol 9666. LNCS. Springer, Heidelberg, pp 327\u2013357"},{"key":"304_CR6","series-title":"LNCS","first-page":"65","volume-title":"ASIACRYPT","author":"B B\u00fcnz","year":"2021","unstructured":"B\u00fcnz B, Maller M, Mishra P, Tyagi N, Vesely P (2021) Proofs for inner pairing products and applications. In: Tibouchi M, Wang H (eds) ASIACRYPT, vol 13092. LNCS. Springer, Cham, pp 65\u201397"},{"key":"304_CR7","doi-asserted-by":"crossref","unstructured":"B\u00fcnz B, Bootle J, Boneh D, Poelstra A, uille P, Maxwell G (2018) Bulletproofs: short proofs for confidential transactions and more. In: S &P, pp 315\u2013334. IEEE, Los Alamitos","DOI":"10.1109\/SP.2018.00020"},{"key":"304_CR8","series-title":"LNCS","first-page":"189","volume-title":"ASIACRYPT","author":"M Campanelli","year":"2022","unstructured":"Campanelli M, Nitulescu A, R\u00e0fols C, Zacharakis A, Zapico A (2022) Linear-map vector commitments and their practical applications. In: Agrawal S, Lin D (eds) ASIACRYPT, vol 13794. LNCS. Springer, Cham, pp 189\u2013219"},{"key":"304_CR9","doi-asserted-by":"publisher","first-page":"42067","DOI":"10.1109\/ACCESS.2022.3167806","volume":"10","author":"H Chung","year":"2022","unstructured":"Chung H, Han K, Ju C, Kim M, Seo JH (2022) Bulletproofs+: Shorter proofs for a privacy-enhanced distributed ledger. IEEE Access 10:42067\u201342082","journal-title":"IEEE Access"},{"key":"304_CR10","series-title":"LNCS","first-page":"527","volume-title":"PKC","author":"V Daza","year":"2020","unstructured":"Daza V, R\u00e0fols C, Zacharakis A (2020) Updateable inner product argument with logarithmic verifier and applications. In: Kiayias A, Kohlweiss M, Wallden P, Zikas V (eds) PKC, vol 12110. LNCS. Springer, Cham, pp 527\u2013557"},{"key":"304_CR11","series-title":"LNCS","first-page":"33","volume-title":"CRYPTO","author":"G Fuchsbauer","year":"2018","unstructured":"Fuchsbauer G, Kiltz E, Loss J (2018) The algebraic group model and its applications. In: Shacham H, Boldyreva A (eds) CRYPTO, vol 10992. LNCS. Springer, Cham, pp 33\u201362"},{"key":"304_CR12","series-title":"LNCS","first-page":"203","volume-title":"FC","author":"N Gailly","year":"2022","unstructured":"Gailly N, Maller M, Nitulescu A (2022) Snarkpack: Practical SNARK aggregation. In: Eyal I, Garay JA (eds) FC, vol 13411. LNCS. Springer, Cham, pp 203\u2013229"},{"key":"304_CR13","series-title":"LNCS","first-page":"192","volume-title":"CRYPTO","author":"J Groth","year":"2009","unstructured":"Groth J (2009) Linear algebra with sub-linear zero-knowledge arguments. In: Halevi S (ed) CRYPTO, vol 5677. LNCS. Springer, Heidelberg, pp 192\u2013208"},{"key":"304_CR14","first-page":"2093","volume-title":"CCS","author":"M Hoffmann","year":"2019","unstructured":"Hoffmann M, Kloo\u00df M, Rupp A (2019) Efficient zero-knowledge arguments in the discrete log setting, revisited. In: Cavallaro L, Kinder J, Wang X, Katz J (eds) CCS. ACM, New York, pp 2093\u20132110"},{"key":"304_CR15","series-title":"LNCS","first-page":"177","volume-title":"ASIACRYPT","author":"A Kate","year":"2010","unstructured":"Kate A, Zaverucha GM, Goldberg I (2010) Constant-size commitments to polynomials and their applications. In: Abe M (ed) ASIACRYPT, vol 6477. LNCS. Springer, Heidelberg, pp 177\u2013194"},{"key":"304_CR16","series-title":"LNCS","first-page":"403","volume-title":"ASIACRYPT","author":"S Kim","year":"2022","unstructured":"Kim S, Lee H, Seo JH (2022) Efficient zero-knowledge arguments in discrete logarithm setting: Sublogarithmic proof or sublinear verifier. In: Agrawal S, Lin D (eds) ASIACRYPT, vol 13792. LNCS. Springer, Cham, pp 403\u2013433"},{"key":"304_CR17","doi-asserted-by":"publisher","first-page":"5332","DOI":"10.1109\/TIFS.2023.3300525","volume":"18","author":"S Kim","year":"2023","unstructured":"Kim S, Lee G, Lee H, Seo JH (2023) Leopard: Sublinear verifier inner product argument under discrete logarithm assumption. IEEE Trans Inf Forens Secur 18:5332\u20135344","journal-title":"IEEE Trans Inf Forens Secur"},{"key":"304_CR18","series-title":"LNCS","first-page":"236","volume-title":"AFRICACRYPT","author":"V Kuchta","year":"2022","unstructured":"Kuchta V, Sahu RA, Sharma G (2022) Lattice-based inner product argument. In: Batina L, Daemen J (eds) AFRICACRYPT, vol 13503. LNCS. Springer, Cham, pp 236\u2013268"},{"key":"304_CR19","first-page":"2057","volume-title":"CCS","author":"RWF Lai","year":"2019","unstructured":"Lai RWF, Malavolta G, Ronge V (2019) Succinct arguments for bilinear group arithmetic: Practical structure-preserving cryptography. In: Cavallaro L, Kinder J, Wang X, Katz J (eds) CCS. ACM, New York, pp 2057\u20132074"},{"key":"304_CR20","series-title":"LNCS","first-page":"1","volume-title":"TCC","author":"J Lee","year":"2021","unstructured":"Lee J (2021) Dory: Efficient, transparent arguments for generalised inner products and polynomial commitments. In: Nissim K, Waters B (eds) TCC, vol 13043. LNCS. Springer, Cham, pp 1\u201334"},{"key":"304_CR21","series-title":"LNCS","first-page":"214","volume-title":"IWSEC","author":"H Lee","year":"2023","unstructured":"Lee H, Seo JH (2023) TENET: sublogarithmic proof and sublinear verifier inner product argument without a trusted setup. In: Shikata J, Kuzuno H (eds) IWSEC, vol 14128. LNCS. Springer, Cham, pp 214\u2013234"},{"key":"304_CR22","series-title":"LNCS","first-page":"129","volume-title":"CRYPTO","author":"TP Pedersen","year":"1991","unstructured":"Pedersen TP (1991) Non-interactive and information-theoretic secure verifiable secret sharing. In: Feigenbaum J (ed) CRYPTO, vol 576. LNCS. Springer, Heidelberg, pp 129\u2013140"},{"key":"304_CR23","doi-asserted-by":"crossref","unstructured":"Wahby RS, Tzialla I, Shelat A, Thaler J, Walfish M (2018) Doubly-efficient zksnarks without trusted setup. In: S &P, pp. 926\u2013943. IEEE, Los Alamitos","DOI":"10.1109\/SP.2018.00060"},{"key":"304_CR24","series-title":"LNCS","first-page":"341","volume-title":"ICICS","author":"Z Zhang","year":"2021","unstructured":"Zhang Z, Zhou Z, Li W, Tao H (2021) An optimized inner product argument with more application scenarios. In: Gao D, Li Q, Guan X, Liao X (eds) ICICS, vol 12919. LNCS. Springer, Cham, pp 341\u2013357"},{"key":"304_CR25","doi-asserted-by":"crossref","unstructured":"Zhang J, Xie T, Zhang Y, Song D (2020) Transparent polynomial delegation and its applications to zero knowledge proof. In: S &P, pp 859\u2013876. IEEE, Los Alamitos","DOI":"10.1109\/SP40000.2020.00052"},{"issue":"3","key":"304_CR26","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1049\/ise2.12111","volume":"17","author":"Z Zhou","year":"2023","unstructured":"Zhou Z, Zhang Z, Tao H, Li T, Zhao B (2023) Efficient inner product arguments and their applications in range proofs. IET Inf Secur 17(3):485\u2013504","journal-title":"IET Inf Secur"}],"container-title":["Cybersecurity"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s42400-024-00304-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1186\/s42400-024-00304-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s42400-024-00304-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,10]],"date-time":"2025-07-10T08:32:27Z","timestamp":1752136347000},"score":1,"resource":{"primary":{"URL":"https:\/\/cybersecurity.springeropen.com\/articles\/10.1186\/s42400-024-00304-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,9]]},"references-count":26,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["304"],"URL":"https:\/\/doi.org\/10.1186\/s42400-024-00304-x","relation":{},"ISSN":["2523-3246"],"issn-type":[{"value":"2523-3246","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,7,9]]},"assertion":[{"value":"17 April 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 June 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 July 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"45"}}