{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T07:05:31Z","timestamp":1784271931272,"version":"3.55.0"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T00:00:00Z","timestamp":1784246400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T00:00:00Z","timestamp":1784246400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"National Cryptologic Science Foundation of China","award":["2025NCSF02028"],"award-info":[{"award-number":["2025NCSF02028"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["3282024001"],"award-info":[{"award-number":["3282024001"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cybersecurity"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Existing threshold Pointcheval\u2013Sanders (PS) signature schemes largely assume equal-weight participants and therefore cannot directly express heterogeneous signing authority in semi-trusted multi-organization settings. This paper presents a dealer-assisted weighted threshold PS signing framework by integrating PS signatures with a Chinese Remainder Theorem (CRT)-based weighted ramp secret sharing (WRSS) mechanism. The main technical contribution is a CRT-to-PS integration method that converts weight-dependent modular shares into exponents while preserving the standard PS verification equation in prime-order bilinear groups. The WRSS base modulus is instantiated as the same prime order used by the bilinear-group scalar field, removing any modular mismatch between CRT reconstruction and PS exponentiation. As a result, the bit-length of each private-key share is proportional to the participant\u2019s weight, and any signing subset whose total weight reaches the reconstruction threshold can jointly generate a valid PS signature. Meanwhile, any coalition whose total weight is at most the privacy threshold learns no information about the master secret beyond what is implied by the public key. We analyze the construction in the semi-honest model and in the WRSS-hybrid model, where the WRSS subroutines are idealized, and establish correctness, unforgeability, and randomization-based non-linkability. We further quantify the incremental computation and communication overhead introduced by the CRT-based weighting layer. The analysis separates CRT arithmetic from the bounded overflow-selection cost and includes heterogeneous-weight parameter tests. The proposed scheme is intended for dealer-assisted, semi-trusted, regulated multi-organization applications, such as joint authorization and credential-issuance workflows with policy-defined participant weights.<\/jats:p>","DOI":"10.1186\/s42400-026-00627-x","type":"journal-article","created":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T06:42:55Z","timestamp":1784270575000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A dealer-assisted weighted threshold Pointcheval\u2013Sanders signature scheme from CRT-based weighted ramp secret sharing"],"prefix":"10.1186","volume":"9","author":[{"given":"Yanshuo","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0005-5023-7203","authenticated-orcid":false,"given":"Jiayin","family":"Kong","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ke","family":"En","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ying","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yatao","family":"Yang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaohong","family":"Qin","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,17]]},"reference":[{"key":"627_CR1","doi-asserted-by":"publisher","first-page":"382","DOI":"10.1007\/978-3-030-88238-9_19","volume-title":"Progress in Cryptology - LATINCRYPT 2021","author":"DF Aranha","year":"2021","unstructured":"Aranha DF, Dalskov A, Escudero D, Orlandi C (2021) Improved threshold signatures, proactive secret sharing, and input certification from lss isomorphisms. In: Longa P, R\u00e0fols C (eds) Progress in Cryptology - LATINCRYPT 2021. Springer, Cham, pp 382\u2013404"},{"issue":"2","key":"627_CR2","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1109\/TIT.1983.1056651","volume":"29","author":"C Asmuth","year":"1983","unstructured":"Asmuth C, Bloom J (1983) A modular approach to key safeguarding. IEEE Trans Inf Theory 29(2):208\u2013210. https:\/\/doi.org\/10.1109\/TIT.1983.1056651","journal-title":"IEEE Trans Inf Theory"},{"key":"627_CR3","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1007\/3-540-39568-7_20","volume-title":"Advances in Cryptology","author":"GR Blakley","year":"1985","unstructured":"Blakley GR, Meadows C (1985) Security of ramp schemes. In: Blakley GR, Chaum D (eds) Advances in Cryptology. Springer, Berlin, Heidelberg, pp 242\u2013268"},{"key":"627_CR4","first-page":"31","volume-title":"Public Key Cryptography \u2013 PKC 2003","author":"A Boldyreva","year":"2002","unstructured":"Boldyreva A (2002) Threshold signatures, multisignatures and blind signatures based on the gap-diffie-hellman-group signature scheme. In: Desmedt YG (ed) Public Key Cryptography \u2013 PKC 2003. Springer, Berlin, Heidelberg, pp 31\u201346"},{"key":"627_CR5","doi-asserted-by":"publisher","first-page":"551","DOI":"10.1007\/978-3-031-15985-5_19","volume-title":"Advances in Cryptology - CRYPTO 2022","author":"D Boneh","year":"2022","unstructured":"Boneh D, Komlo C (2022) Threshold signatures with private accountability. In: Dodis Y, Shrimpton T (eds) Advances in Cryptology - CRYPTO 2022. Springer, Cham, pp 551\u2013581"},{"key":"627_CR6","doi-asserted-by":"publisher","first-page":"514","DOI":"10.1007\/3-540-45682-1_30","volume-title":"Advances in Cryptology \u2013 ASIACRYPT 2001","author":"D Boneh","year":"2001","unstructured":"Boneh D, Lynn B, Shacham H (2001) Short signatures from the weil pairing. In: Boyd C (ed) Advances in Cryptology \u2013 ASIACRYPT 2001. Springer, Berlin, Heidelberg, pp 514\u2013532"},{"key":"627_CR7","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1007\/978-3-540-28628-8_4","volume-title":"Advances in Cryptology - CRYPTO 2004","author":"J Camenisch","year":"2004","unstructured":"Camenisch J, Lysyanskaya A (2004) Signature schemes and anonymous credentials from bilinear maps. In: Franklin M (ed) Advances in Cryptology - CRYPTO 2004. Springer, Berlin, Heidelberg, pp 56\u201372"},{"key":"627_CR8","doi-asserted-by":"publisher","first-page":"348","DOI":"10.1007\/978-981-99-8724-5_11","volume-title":"Advances in Cryptology - ASIACRYPT 2023","author":"E Crites","year":"2023","unstructured":"Crites E, Kohlweiss M, Preneel B, Sedaghat M, Slamanig D (2023) Threshold structure-preserving signatures. In: Guo J, Steinfeld R (eds) Advances in Cryptology - ASIACRYPT 2023. Springer, Singapore, pp 348\u2013382"},{"key":"627_CR9","doi-asserted-by":"publisher","first-page":"314","DOI":"10.1007\/978-3-031-91098-2_12","volume-title":"Advances in Cryptology - EUROCRYPT 2025","author":"S Das","year":"2025","unstructured":"Das S, Pinkas B, Tomescu A, Xiang Z (2025) Distributed randomness using weighted vufs. In: Fehr S, Fouque P-A (eds) Advances in Cryptology - EUROCRYPT 2025. Springer, Cham, pp 314\u2013344"},{"key":"627_CR10","doi-asserted-by":"publisher","unstructured":"Das S, Camacho P, Xiang Z, Nieto J, B\u00fcnz B, Ren L (2023) Threshold signatures from inner product argument: Succinct, weighted, and multi-threshold. In: Proceedings of the 2023 ACM SIGSAC Conference on Computer and Communications Security, pp. 356\u2013370. Association for Computing Machinery, New York, NY, USA. https:\/\/doi.org\/10.1145\/3576915.3623096","DOI":"10.1145\/3576915.3623096"},{"key":"627_CR11","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1007\/0-387-34805-0_28","volume-title":"Advances in Cryptology \u2013 CRYPTO\u2019 89 Proceedings","author":"Y Desmedt","year":"1990","unstructured":"Desmedt Y, Frankel Y (1990) Threshold cryptosystems. In: Brassard G (ed) Advances in Cryptology \u2013 CRYPTO\u2019 89 Proceedings. Springer, New York, NY, pp 307\u2013315"},{"key":"627_CR12","doi-asserted-by":"publisher","unstructured":"Feldman P (1987) A practical scheme for non-interactive verifiable secret sharing. In: 28th Annual Symposium on Foundations of Computer Science (SFCS 1987), pp. 427\u2013438. https:\/\/doi.org\/10.1109\/SFCS.1987.4","DOI":"10.1109\/SFCS.1987.4"},{"key":"627_CR13","doi-asserted-by":"publisher","first-page":"295","DOI":"10.1007\/978-3-031-38557-5_10","volume-title":"Advances in Cryptology - CRYPTO 2023","author":"S Garg","year":"2023","unstructured":"Garg S, Jain A, Mukherjee P, Sinha R, Wang M, Zhang Y (2023) Cryptography with weights: Mpc, encryption and signatures. In: Handschuh H, Lysyanskaya A (eds) Advances in Cryptology - CRYPTO 2023. Springer, Cham, pp 295\u2013327"},{"key":"627_CR14","doi-asserted-by":"crossref","unstructured":"Garg S, Jain A, Mukherjee P, Sinha R, Wang M, Zhang Y (2024) hints: Threshold signatures with silent setup. 2024 IEEE Symposium on Security and Privacy (SP). pp 3034\u20133052","DOI":"10.1109\/SP54263.2024.00057"},{"key":"627_CR15","first-page":"161","volume":"15","author":"S Iftene","year":"2005","unstructured":"Iftene S, Boureanu I (2005) Weighted threshold secret sharing based on the chinese remainder theorem. Scientific Annals of Cuza University 15:161\u2013172","journal-title":"Scientific Annals of Cuza University"},{"issue":"9","key":"627_CR16","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1002\/ecjc.4430720906","volume":"72","author":"M Ito","year":"1989","unstructured":"Ito M, Saito A, Nishizeki T (1989) Secret sharing scheme realizing general access structure. Electron Commun in Japan (Part III Fundamental Electronic Science) 72(9):56\u201364. https:\/\/doi.org\/10.1002\/ecjc.4430720906","journal-title":"Electron Commun in Japan (Part III Fundamental Electronic Science)"},{"key":"627_CR17","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1007\/978-3-030-81652-0_2","volume-title":"Selected Areas in Cryptography","author":"C Komlo","year":"2021","unstructured":"Komlo C, Goldberg I (2021) Frost: Flexible round-optimized schnorr threshold signatures. In: Dunkelman O, Jacobson MJ Jr, O\u2019Flynn C (eds) Selected Areas in Cryptography. Springer, Cham, pp 34\u201365"},{"key":"627_CR18","doi-asserted-by":"publisher","first-page":"2217","DOI":"10.1109\/TIFS.2023.3347968","volume":"19","author":"M Li","year":"2024","unstructured":"Li M, Ding H, Wang Q, Zhang M, Meng W, Zhu L, Zhang Z, Lin X (2024) Decentralized threshold signatures with dynamically private accountability. IEEE Trans Inf Forensics Secur 19:2217\u20132230. https:\/\/doi.org\/10.1109\/TIFS.2023.3347968","journal-title":"IEEE Trans Inf Forensics Secur"},{"key":"627_CR19","doi-asserted-by":"publisher","unstructured":"Lin., Liao Y, Zeng W, Wei Y, Chen D, Yuan X, Li Y, Erkan U, Toktas A, Zhang C, Zhang Y, Gao S (2026) 3D non-degenerate hyperchaos: Design, analysis, and application in image encryption. IEEE Transactions on Consumer Electronics PP(99), 1\u20131 https:\/\/doi.org\/10.1109\/TCE.2026.3672135","DOI":"10.1109\/TCE.2026.3672135"},{"issue":"6","key":"627_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3808699","volume":"22","author":"Y Lin","year":"2026","unstructured":"Lin Y, Wei Y, Chen D, Li Y, Erkan U, Toktas A, Gao S, Zhang Y (2026) Cryptanalysis and improvement of a video cryptosystem via chaos and S-Box. ACM Trans Multimed Comput Commun Appl 22(6):1\u201328. https:\/\/doi.org\/10.1145\/3808699","journal-title":"ACM Trans Multimed Comput Commun Appl"},{"key":"627_CR21","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1007\/978-3-319-57048-8_6","volume-title":"Tutorials on the Foundations of Cryptography: Dedicated to Oded Goldreich","author":"Y Lindell","year":"2017","unstructured":"Lindell Y (2017) How to simulate it - a tutorial on the simulation proof technique. In: Lindell Y (ed) Tutorials on the Foundations of Cryptography: Dedicated to Oded Goldreich. Springer, Cham, pp 277\u2013346"},{"key":"627_CR22","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1007\/3-540-39466-4_27","volume-title":"Cryptography","author":"M Mignotte","year":"1983","unstructured":"Mignotte M (1983) How to share a secret. In: Beth T (ed) Cryptography. Springer, Berlin, Heidelberg, pp 371\u2013375"},{"key":"627_CR23","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1007\/3-540-46766-1_9","volume-title":"Advances in Cryptology \u2013 CRYPTO \u201991","author":"TP Pedersen","year":"1992","unstructured":"Pedersen TP (1992) Non-interactive and information-theoretic secure verifiable secret sharing. In: Feigenbaum J (ed) Advances in Cryptology \u2013 CRYPTO \u201991. Springer, Berlin, Heidelberg, pp 129\u2013140"},{"key":"627_CR24","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1007\/978-3-319-29485-8_7","volume-title":"Topics in Cryptology - CT-RSA 2016","author":"D Pointcheval","year":"2016","unstructured":"Pointcheval D, Sanders O (2016) Short randomizable signatures. In: Sako K (ed) Topics in Cryptology - CT-RSA 2016. Springer, Cham, pp 111\u2013126"},{"key":"627_CR25","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1007\/BF00196725","volume":"4","author":"CP Schnorr","year":"1991","unstructured":"Schnorr CP (1991) Efficient signature generation by smart cards. J Cryptol 4:161\u2013174. https:\/\/doi.org\/10.1007\/BF00196725","journal-title":"J Cryptol"},{"issue":"11","key":"627_CR26","doi-asserted-by":"publisher","first-page":"612","DOI":"10.1145\/359168.359176","volume":"22","author":"A Shamir","year":"1979","unstructured":"Shamir A (1979) How to share a secret. Commun ACM 22(11):612\u2013613. https:\/\/doi.org\/10.1145\/359168.359176","journal-title":"Commun ACM"},{"key":"627_CR27","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1007\/3-540-45539-6_15","volume-title":"Advances in Cryptology \u2013 EUROCRYPT 2000","author":"V Shoup","year":"2000","unstructured":"Shoup V (2000) Practical threshold signatures. In: Preneel B (ed) Advances in Cryptology \u2013 EUROCRYPT 2000. Springer, Berlin, Heidelberg, pp 207\u2013220"},{"key":"627_CR28","doi-asserted-by":"publisher","unstructured":"Sonnino A, Al-Bassam M, Bano S, Meiklejohn S, Danezis G (2019) Coconut: Threshold issuance selective disclosure credentials with applications to distributed ledgers. In: 26th Annual Network and Distributed System Security Symposium (NDSS 2019). The Internet Society, San Diego, CA, USA. https:\/\/doi.org\/10.14722\/ndss.2019.23272. arXiv:1802.07344","DOI":"10.14722\/ndss.2019.23272"},{"issue":"8","key":"627_CR29","doi-asserted-by":"publisher","first-page":"3883","DOI":"10.13328\/j.cnki.jos.007255","volume":"36","author":"CH Tang","year":"2025","unstructured":"Tang CH, Zhao YQ, Yang XY, Feng Q, Yu Y (2025) Weighted threshold sm2 signature scheme. J Soft 36(8):3883\u20133895. https:\/\/doi.org\/10.13328\/j.cnki.jos.007255","journal-title":"J Soft"},{"key":"627_CR30","unstructured":"Waters B, Wu DJ (2025) Silent Threshold Cryptography from Pairings: Expressive Policies in the Plain Model. Cryptology ePrint Archive, Paper 2025\/1547. https:\/\/eprint.iacr.org\/2025\/1547"},{"key":"627_CR31","doi-asserted-by":"publisher","first-page":"7033","DOI":"10.1109\/TIFS.2024.3428848","volume":"19","author":"Y Xie","year":"2024","unstructured":"Xie Y, Fan Q, Zhang C, Wu T, Zhou Y, He D, Zhu L (2024) Accountable and secure threshold eddsa signature and its applications. IEEE Trans Inf Forensics Secur 19:7033\u20137046. https:\/\/doi.org\/10.1109\/TIFS.2024.3428848","journal-title":"IEEE Trans Inf Forensics Secur"},{"issue":"2","key":"627_CR32","doi-asserted-by":"publisher","first-page":"3738","DOI":"10.1109\/TDSC.2025.3640211","volume":"23","author":"X Zhang","year":"2026","unstructured":"Zhang X, Zhou J, Cao Z, Dong X, Ning J (2026) Weighted threshold anonymous credentials with redactable fine-grained blind signature for auditable lending system in blockchains. IEEE Trans Dependable Secure Comput 23(2):3738\u20133752. https:\/\/doi.org\/10.1109\/TDSC.2025.3640211","journal-title":"IEEE Trans Dependable Secure Comput"}],"container-title":["Cybersecurity"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s42400-026-00627-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1186\/s42400-026-00627-x","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s42400-026-00627-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T06:42:58Z","timestamp":1784270578000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1186\/s42400-026-00627-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,17]]},"references-count":32,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,12]]}},"alternative-id":["627"],"URL":"https:\/\/doi.org\/10.1186\/s42400-026-00627-x","relation":{},"ISSN":["2523-3246"],"issn-type":[{"value":"2523-3246","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,17]]},"assertion":[{"value":"28 April 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 July 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 July 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no competing interest.","order":1,"name":"Ethics","label":"Competing interest","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"192"}}