{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,29]],"date-time":"2026-01-29T21:47:07Z","timestamp":1769723227217,"version":"3.49.0"},"reference-count":25,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,2,25]],"date-time":"2022-02-25T00:00:00Z","timestamp":1645747200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Zhejiang Province of China","award":["LZ17F020002"],"award-info":[{"award-number":["LZ17F020002"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Zero-Knowledge Proof is widely used in blockchains. For example, zk-SNARK is used in Zcash as its core technology to identifying transactions without the exposure of the actual transaction values. Up to now, various range proofs have been proposed, and their efficiency and range-flexibility have also been improved. Bootle et al. used the inner product method and recursion to construct an efficient Zero-Knowledge Proof in 2016. Later, Benediky B\u00fcnz et al. proposed an efficient range proof scheme called Bulletproofs, which can convince the verifier that a secret number lies in [0,2\u03ba\u22121] with \u03ba being a positive integer. By combining the inner-product and Lagrange\u2019s four-square theorem, we propose a range proof scheme called Cuproof. Our Cuproof can make a range proof to show that a secret number v lies in an interval [a,b] with no exposure of the real value v or other extra information leakage about v. It is a good and practical method to protect privacy and information security. In Bulletproofs, the communication cost is 6+2log\u03ba, while in our Cuproof, all the communication cost, the proving time and the verification time are of constant sizes.<\/jats:p>","DOI":"10.3390\/e24030334","type":"journal-article","created":{"date-parts":[[2022,2,27]],"date-time":"2022-02-27T20:46:17Z","timestamp":1645994777000},"page":"334","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Cuproof: Range Proof with Constant Size"],"prefix":"10.3390","volume":"24","author":[{"given":"Cong","family":"Deng","sequence":"first","affiliation":[{"name":"School of Communication Engineering, Hangzhou Dianzi University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2049-8961","authenticated-orcid":false,"given":"Lin","family":"You","sequence":"additional","affiliation":[{"name":"School of Cyberspace Security, Hangzhou Dianzi University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xianghong","family":"Tang","sequence":"additional","affiliation":[{"name":"School of Communication Engineering, Hangzhou Dianzi University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3061-2388","authenticated-orcid":false,"given":"Gengran","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Cyberspace Security, Hangzhou Dianzi University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shuhong","family":"Gao","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1287","DOI":"10.1108\/JEIM-10-2020-0395","article-title":"Exploring the intellectual cores of the blockchain\u2013Internet of Things (BIoT)","volume":"34","author":"Tsang","year":"2021","journal-title":"J. Enterp. Inf. Manag."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Fedorov, I.R., Pimenov, A.V., Panin, G.A., and Bezzateev, S.V. (June, January 31). Blockchain in 5G Networks: Perfomance Evaluation of Private Blockchain. Proceedings of the 2021 Wave Electronics and its Application in Information and Telecommunication Systems (WECONF), St. Petersburg, Russia.","DOI":"10.1109\/WECONF51603.2021.9470519"},{"key":"ref_3","unstructured":"Nakamoto, S. (2022, February 21). Bitcoin: A Peer-to-Peer Electronic Cash System. Available online: http:\/\/www.bitcoin.org\/bitcoin.pdf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1109\/MNET.011.2000473","article-title":"A survey on zero-knowledge proof in blockchain","volume":"35","author":"Sun","year":"2021","journal-title":"IEEE Netw."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"B\u00fcnz, B., Bootle, J., Boneh, D., Poelstra, A., Wuille, P., and Maxwell, G. (2018, January 20\u201324). Bulletproofs: Short Proofs for Confidential Transactions and More. Proceedings of the 2018 IEEE Symposium on Security and Privacy, San Francisco, CA, USA.","DOI":"10.1109\/SP.2018.00020"},{"key":"ref_6","unstructured":"Pomerance, C. (1987). Gradual and Verifiable Release of a Secret (Extended Abstract). Advances in Cryptology\u2014CRYPTO \u201987, Springer."},{"key":"ref_7","unstructured":"Gabizon, A., Williamson, Z.J., and Ciobotaru, O. (2021, December 08). PLONK: Permutations over Lagrange-Bases for Oecumenical Noninteractive Arguments of Knowledge. Available online: https:\/\/eprint.iacr.org\/2019\/953."},{"key":"ref_8","unstructured":"Maller, M., Bowe, S., Kohlweiss, M., and Meiklejohn, S. (2021, December 08). Sonic: Zero-Knowledge SNARKs from Linear-Size Universal and Updateable Structured Reference Strings. Available online: https:\/\/eprint.iacr.org\/2019\/099."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1145\/3421473.3421479","article-title":"Verifiable state machines: Proofs that untrusted services operate correctly","volume":"54","author":"Setty","year":"2020","journal-title":"ACM SIGOPS Oper. Syst. Rev."},{"key":"ref_10","unstructured":"Ben-Sasson, E., Bentov, I., Horesh, Y., and Riabzev, M. (2021, August 15). Scalable, Transparent, and Post-Quantum Secure Computational Integrity. Available online: https:\/\/eprint.iacr.org\/2018\/046."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ben Sasson, E., Chiesa, A., Garman, C., Green, M., Miers, I., Tromer, E., and Virza, M. (2014, January 18\u201321). Zerocash: Decentralized Anonymous Payments from Bitcoin. Proceedings of the 2014 IEEE Symposium on Security and Privacy, San Jose, CA, USA.","DOI":"10.1109\/SP.2014.36"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"474","DOI":"10.1080\/17517575.2019.1644672","article-title":"How does overconfidence affect information security investment and information security performance?","volume":"15","author":"Dong","year":"2021","journal-title":"J. Enterp. Inf. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Nyberg, K. (1998). Easy come\u2014Easy go divisible cash. Advances in Cryptology\u2014EUROCRYPT\u201998, Springer.","DOI":"10.1007\/BFb0054112"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Preneel, B. (2000). Efficient Proofs that a Committed Number Lies in an Interval. Advances in Cryptology\u2014EUROCRYPT 2000, Springer.","DOI":"10.1007\/3-540-45539-6"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1002\/cpa.3160390713","article-title":"Randomized algorithms in number theory","volume":"39","author":"Rabin","year":"1986","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Laih, C.S. (2003). On Diophantine Complexity and Statistical Zero-Knowledge Arguments. Advances in Cryptology\u2014ASIACRYPT 2003, Springer.","DOI":"10.1007\/b94617"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Ioannidis, J., Keromytis, A., and Yung, M. (2005). Non-interactive Zero-Knowledge Arguments for Voting. Applied Cryptography and Network Security, Springer.","DOI":"10.1007\/b137093"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Cachin, C., and Camenisch, J.L. (2004). Short Signatures Without Random Oracles. Advances in Cryptology\u2014EUROCRYPT 2004, Springer.","DOI":"10.1007\/b97182"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Yung, M., Dodis, Y., Kiayias, A., and Malkin, T. (2006). k-Times Anonymous Authentication with a Constant Proving Cost. Public Key Cryptography\u2014PKC 2006, Springer.","DOI":"10.1007\/11745853"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Pieprzyk, J. (2008). Efficient Protocols for Set Membership and Range Proofs. Advances in Cryptology\u2014ASIACRYPT 2008, Springer.","DOI":"10.1007\/978-3-540-89255-7"},{"key":"ref_21","unstructured":"Belenkiy, M. (2021, September 05). U-Prove Range Proof Extension. Available online: https:\/\/www.microsoft.com\/en-us\/research\/publication\/u-prove-range-proof-extension\/."},{"key":"ref_22","unstructured":"Paquin, C., and Zaverucha, G. (2021, December 08). U-Prove Cryptographic Specification V1.1 (Revision 3). Available online: https:\/\/www.microsoft.com\/en-us\/research\/publication\/u-prove-cryptographic-specification-v1-1-revision-3\/."},{"key":"ref_23","unstructured":"Fischlin, M., and Coron, J.S. (2016). Efficient Zero-Knowledge Arguments for Arithmetic Circuits in the Discrete Log Setting. Advances in Cryptology\u2014EUROCRYPT 2016, Springer."},{"key":"ref_24","unstructured":"Fischlin, M., and Coron, J.S. (2016). On the Size of Pairing-Based Non-interactive Arguments. Advances in Cryptology\u2014EUROCRYPT 2016, Springer."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Li, K., Yang, R., Au, M.H., and Xu, Q. (2017, January 6\u20138). Practical range proof for cryptocurrency monero with provable security. Proceedings of the International Conference on Information and Communications Security, Beijing, China.","DOI":"10.1007\/978-3-319-89500-0_23"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/334\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:27:54Z","timestamp":1760135274000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/3\/334"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,25]]},"references-count":25,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["e24030334"],"URL":"https:\/\/doi.org\/10.3390\/e24030334","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2,25]]}}}