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Bulletproofs: Short Proofs for Confidential Transactions and More
Open accessAbstract
We propose Bulletproofs, a new non-interactive zero-knowledge proof protocol with very short proofs and without a trusted setup; the proof size is only logarithmic in the witness size. Bulletproofs are especially well suited for efficient range proofs on committed values: they enable proving that a committed value is in a range using only 2 log_2(n)+9 group and field elements, where n is the bit length of the range. Proof generation and verification times are linear in n. Bulletproofs greatly improve on the linear (in n) sized range proofs in existing proposals for confidential transactions in Bitcoin and other cryptocurrencies. Moreover, Bulletproofs supports aggregation of range proofs, so that a party can prove that m commitments lie in a given range by providing only an additive O(log(m)) group elements over the length of a single proof. To aggregate proofs from multiple parties, we enable the parties to generate a single proof without revealing their inputs to each other via a simple multi-party computation (MPC) protocol for constructing Bulletproofs. This MPC protocol uses either a constant number of rounds and linear communication, or a logarithmic number of rounds and logarithmic communication. We show that verification time, while asymptotically linear, is very efficient in practice. The marginal cost of batch verifying 32 aggregated range proofs is less than the cost of verifying 32 ECDSA signatures. Bulletproofs build on the techniques of Bootle et al. (EUROCRYPT 2016). Beyond range proofs, Bulletproofs provide short zero-knowledge proofs for general arithmetic circuits while only relying on the discrete logarithm assumption and without requiring a trusted setup. We discuss many applications that would benefit from Bulletproofs, primarily in the area of cryptocurrencies. The efficiency of Bulletproofs is particularly well suited for the distributed and trustless nature of blockchains. The full version of this article is available on ePrint.
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“The B ulletproofs p rotocol by B¨ unz et al. [130] has contributed significantly to the progress of ZKP.”
From A Survey of Self-Sovereign Identity Ecosystem (2021) · cites this paper as [130]
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“In this sense, Key-Policy Attribute-Based Encryption (KP-ABE) [129] or Ciphertext-Policy Attribute-Based Encryption (CP-ABE) [130], which allows to de ne access control policies in the encryption itself, allows that only the users with the right attributes can decrypt the data.”
From Privacy-Preserving Solutions for Blockchain: Review and Challenges (2019) · cites this paper as [130]
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Cites
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Items 801–900 of 1,000
Fully-Adaptive Two-Round Threshold Schnorr Signatures from DDH
Gerhart, P., Calsi, D. L., Russo, L., & Schröder, D. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-25317-0_9 · Cited by 1
“To keep the proof system transparent and avoid additional assumptions, we can leverage a (commit-and-prove) simulation-extractable scheme based on Bulletproofs [BBB + 18], which yields logarithmic proof size in the polynomial degree d under the discrete logarithm assumption.”
Soloist: Distributed SNARK for R1CS with Constant Proof Size
Li, W., Zhang, Z., Li, Y., Zhu, P., Hong, C., & Liu, J. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-25336-1_3 · Cited by 1
Revisiting Linkable Ring Signatures with Logarithmic Verification Complexity
Balla, D., & Chaidos, P. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-26737-5_17 · Cited by 1
“One example is DualRing [26] by Yuen et al, a ring signature scheme with logarithmic size and linear prover time, achieved by compressing the signature using Bulletproofs [7].”
InfluentialSitaula, A., Uddin, M. A., Ayoade, J., Chu, N. H., & Rafeh, R. (2026) · Blockchains · doi:10.3390/blockchains4020005 · Cited by 1
“Alternative constructions, including zk-STARKs and Bulletproofs, have been proposed to address limitations related to trusted setup and scalability, offering improved transparency and post-quantum resilience [25,26].”
Siim, J. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-26734-4_15 · Cited by 1
Griffy, S., Kohlweiss, M., Lysyanskaya, A., & Sengupta, M. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-26740-5_8 · Cited by 1
HyperFond: A Transparent and Post-Quantum Distributed SNARK with Polylogarithmic Communication
Yu, Y., Liu, M., Zhang, Y., Ma, T., Sun, S.-F., Au, M. H., & Gu, D. (2026) · ACM Asia Conference on Computer and Communications Security (ASIA CCS) · doi:10.1145/3779208.3805973 · Cited by 1
“…studies have been carried out on SNARK constructions ([2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and references therein), focusing on reducing reliance on trusted setups [4, 6, 9, 10, 13], achieving post-quantum security [2, 3, 6, 9, 13, 14], and optimizing performance [5, 7, 8, 11, 12].”
SoK: Understanding zkVM: From Research to Practice
Yang, G., Cheng, Y., Tang, H., Tang, H., Zhang, B., & Ren, K. (2026) · ACM Asia Conference on Computer and Communications Security (ASIA CCS) · doi:10.1145/3779208.3805964 · Cited by 1
“Efficient schemes [20, 26] can reduce the number of openings and simplify verification, while more complex constructions achieve smaller proofs but require heavier computation.”
Hopinzero: zkSNARK-Verified Privacy for On-Chain Options with Pooled Liquidity
Košťál, K., Bitarovský, R., & Mastilak, L. (2026) · IEEE International Conference on Blockchain and Cryptocurrency (ICBC) · doi:10.1109/icbc67748.2026.11575488 · Cited by 1
“Bulletproofs remove trusted setup but have more expensive verification [8], [9].”
Strong existential unforgeability and BUFF securities of MPC-in-the-head signatures
Kulkarni, M., & Xagawa, K. (2026) · Designs Codes and Cryptography · doi:10.1007/s10623-026-01851-9 · Cited by 1
Llambí-Morillas, M., & Fernández-Fernández, D. (2026) · Frontiers in Computer Science · doi:10.3389/fcomp.2026.1966725 · Cited by 4
“Groth16 [9] established the basis for efficient pairing-based non-interactive arguments; STARKs [10] removed trusted-setup requirements at the cost of larger proofs; Bulletproofs [11] provided efficient range proofs without trusted setup.”
On the Fiat–Shamir Security of Succinct Arguments from Functional Commitments
Chiesa, A., Guan, Z., Knabenhans, C., & Yu, Z. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-35424-2_7 · Cited by 1
Chopin: Optimal Pairing-Based Multilinear Polynomial Commitments from Bivariate KZG
Belohorec, J., Hubáček, P., Kalsta, A., & Mašková, K. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-35424-2_15 · Cited by 1
Performance Trade-offs in Design of MimbleWimble Proofs of Reserves
Bagad, S., & Vijayakumaran, S. (2020) · IEEE European Symposium on Security and Privacy Workshops (EuroS&PW) · doi:10.1109/eurospw51379.2020.00057 · Cited by 1
“It relies on Bulletproofs [10] to achieve this size reduction.”
MethodInfluentialPisces: A New Zero-Knowledge Protocol for Blockchain Privacy
Fu, S., & Gong, G. (2021) · Lecture notes in computer science · doi:10.1007/978-3-030-70881-8_12 · Cited by 1
Attema, T., Cramer, R., & Rambaud, M. (2021) · TNO Repository · Cited by 1
Combinatorial Spectrum E-Auction for 5G Heterogeneous Networks: A Zether-Based Approach
Zhao, Z., Ying, Z., Cai, Z., & Ma, J. (2021) · Security and Communication Networks · doi:10.1155/2021/1360560 · Cited by 1
An Improved Range Proof with Base-3 Construction
Günsay, E., Onur, C. B., & Cenk, M. (2021) · International Conference on Security of Information and Networks (SIN) · doi:10.1109/sin54109.2021.9699258 · Cited by 1
“[7] propose range proof scheme ‘Bulletproofs’ based on discrete logarithm in elliptic curve.”
BackgroundTrustworthy Blockchain Data Sharing Model Based on Multi-authority ABE and zk-SNARK
Yang, L., Zhang, G., Zhu, J., Wang, X., & Wang, Y. (2021) · Communications in computer and information science · doi:10.1007/978-981-16-7993-3_24 · Cited by 1
One Bad Apple Spoils the Bunch: Transaction DoS in MimbleWimble Blockchains
Tabatabaee, S. A., Nicer, C., Beschastnikh, I., & Feng, C. (2022) · IEEE International Conference on Blockchain and Cryptocurrency (ICBC) · doi:10.1109/icbc54727.2022.9805529 · Cited by 1
Efficient Public Verification of Confidential Supply-Chain Transactions
Becher, K., Schafer, M., Schröpfer, A., & Strufe, T. (2022) · IEEE Conference on Communications and Network Security · doi:10.1109/cns56114.2022.9947231 · Cited by 1
“3 times less storage than Bulletproofs.”
BackgroundResultInfluentialResumable Zero-Knowledge for Circuits from Symmetric Key Primitives
Zhang, H., Wei, P., Xue, H., Deng, Y., Li, J., Wang, W., & Liu, G. (2022) · Lecture notes in computer science · doi:10.1007/978-3-031-22301-3_19 · Cited by 1
“To construct efficient ZK proofs for a larger class of languages, many works focus on ZK proofs for arbitrary circuits [44,11,38,53,13,25,16,45,19,59,10,12], which have relatively short proofs size and verification time.”
MethodA Universally Composable Non-interactive Aggregate Cash System
Jia, Y., Shi-Feng, S., Zhou, H.-S., & Gu, D. (2022) · Lecture notes in computer science · doi:10.1007/978-3-031-22963-3_25 · Cited by 1
The Data Exchange Protocol over Multi-chain Blockchain Using Zero-Knowledge Proof
Li, A., D’Angelo, G., & Tang, S.-K. (2023) · Lecture notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering · doi:10.1007/978-3-031-28990-3_2 · Cited by 1
Zero-Knowledge Arguments for Subverted RSA Groups
Kolonelos, D., Maller, M., & Volkhov, M. (2023) · Lecture notes in computer science · doi:10.1007/978-3-031-31371-4_18 · Cited by 1
“Afterwards one can use a range proof protocol in the prime order group [17, 26] to prove the range of w .”
Dumas, J., Maignan, A., Pernet, C., & Roche, D. S. (2023) · Proceedings on Privacy Enhancing Technologies · doi:10.56553/popets-2023-0085 · Cited by 1
D-KODE: Distributed Mechanism to Manage a Billion Discrete-log Keys
Mangipudi, E. V., & Kate, A. (2022) · ACM Conference on Advances in Financial Technologies (AFT) · doi:10.1145/3558535.3559788 · Cited by 1
“For proving the correct evaluation of z = F (X,k), the server computes z′ = F (X,k′) and r = pw − qz mod τ , r ′ = pw ′ − qz′ mod τ ; forwards the values c = gh ′ and z′ = F (X,k′) The server also computes and forwards zero-knowledge range proof [40] πr , πk proving that r < q, k < u · q such thatw + kq = ∑ui=1 αihi .”
HSchain: Anonymous Permissioned Blockchain with Enhanced Auditability
Hou, J., Xu, L., Zhu, L., Jiang, P., & Song, S. (2023) · IEEE International Conference on Cyber Security and Resilience (CSR) · doi:10.1109/csr57506.2023.10225006 · Cited by 1
“The proof π is a range proof which is generated via the way described in bulletproof [20].”
MethodEfficient Zero-Knowledge Arguments for Matrix Relations over Rings and Non-malleable Enhancement
Tian, Y., Tian, X., & Pang, Y. (2023) · Lecture notes in computer science · doi:10.1007/978-3-031-45513-1_1 · Cited by 1
Implementation and preliminary evaluation of an auditable confidentiality mechanism for DeFi
Li, A., Tang, S.-K., & D’Angelo, G. (2023) · IEEE International Conference on Distributed Computing Systems Workshops (ICDCSW) · doi:10.1109/icdcsw60045.2023.00015 · Cited by 1
“Another Bitcoin-based blockchain that protects anonymity is Monero [7], although unlike Zerocash, Monero makes use of ring signatures and range-proofs [12].”
BackgroundVerifiable Computing in Avionics for Assuring Computer-Integrity without Replication
Reinhart, J., Luettig, B., Huber, N., Liedtke, J., & Annighoefer, B. (2023) · IEEE/AIAA Digital Avionics Systems Conference (DASC) · doi:10.1109/dasc58513.2023.10311290 · Cited by 1
“Newer systems include Bulletproofs [14], Ligero [15], STARK [16], Plonk [17], and Halo [18].”
Methodzk-SNARKs from Codes with Rank Metrics
Do, X.-T., Mac, D.-T., & Vu, Q.-H. (2023) · Lecture notes in computer science · doi:10.1007/978-3-031-47818-5_6 · Cited by 1
“They are an essential component of many privacy-preserving cryptographic systems, including credentials and digital currencies [BG90, CL01, BCC + 09, CKL + 16, FHS19] as well as group signatures [Cv91, BBS04, DP06, BCC + 16] and verifiable computation [GGP10, GGPR13, BCG + 18].”
BackgroundArmed with Faster Crypto: Optimizing Elliptic Curve Cryptography for ARM Processors
Smet, R. D., Blancquaert, R., Godden, T., Steenhaut, K., & Braeken, A. (2024) · Sensors · doi:10.3390/s24031030 · Cited by 1
“In some applications—for example, zero-knowledge proofs (e.g., Bulletproofs [4])—thousands of ECC operations need to be carried out as efficiently as possible.”
BackgroundEfficient Inner-Product Argument from Compressed $$\Sigma $$-Protocols and Applications
Scala, E., & Mostarda, L. (2024) · Lecture notes on data engineering and communications technologies · doi:10.1007/978-3-031-57916-5_28 · Cited by 1
Succinct Verification of Compressed Sigma Protocols in the Updatable SRS Setting
Dutta, M., Ganesh, C., & Jawalkar, N. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-57722-2_10 · Cited by 1
“Introduced by Groth et al. [GKM + 18], the updatable universal structured reference string (SRS) enables parties to update the parameters, while retaining computational soundness against any probabilistic-polynomial time adversary, as long as at least one honest update is performed.”
BackgroundMethodPublic-Coin, Complexity-Preserving, Succinct Arguments of Knowledge for NP from Collision-Resistance
Freitag, C., Paneth, O., & Pass, R. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-58737-5_5 · Cited by 1
Cross-Chain Trusted Information Match Scheme with Privacy-Preserving and Auditability
Chen, Z. D., Lu, Z., & Chen, J. (2024) · Lecture notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering · doi:10.1007/978-3-031-60037-1_6 · Cited by 1
Privacy-Preserving Control of Partitioned Energy Resources
Laufer, E., Levis, P. A., & Rajagopal, R. (2024) · ACM International Conference on Future Energy Systems (e-Energy) · doi:10.1145/3632775.3661988 · Cited by 1
A Batch Payment Scheme with Denomination Privacy
Zhao, L., Zhong, L., Wang, Y., & Zhang, J. (2024) · International Conference on Computer Supported Cooperative Work in Design (CSCWD) · doi:10.1109/cscwd61410.2024.10580360 · Cited by 1
“…batch payment scheme incorporates three types of zero-knowledge proofs to validate the correctness of ciphertexts and transactions: a Sigma proof [26], [27] to prevent overspending, a batch proof for correct ciphertext generation, and a range proof [28] to verify that all payments are non-negative.”
BackgroundResearch on transaction privacy protection solutions for cross‐border commerce
Ma, J., & Li, F. (2024) · IET Blockchain · doi:10.1049/blc2.12080 · Cited by 1
eSTARK: extending STARKs with arguments
Ardévol, H. M., Baylina, J., Guzmán-Albiol, M., & Muñoz, J. L. (2024) · Designs Codes and Cryptography · doi:10.1007/s10623-024-01457-z · Cited by 1
“Compared with completeness and soundness, the zero-knowledge property guarantees the privacy of the prover’s secret information against malicious verifiers.”
A Formal Analysis of the Mimblewimble Cryptocurrency Protocol with a Security Approach
Silveira, A., Betarte, G., & Luna, C. (2024) · CLEI electronic journal · doi:10.19153/cleiej.27.3.4 · Cited by 1
“A non-interactive commitment scheme [31] can be defined as follows: Definition 1 (Non-interactive Commitment Scheme) A non-interactive commitment scheme ζ ( Setup , Com ) consists of two probabilistic polynomial time algorithms, Setup and Com, such that: • Setup generates public parameters for the…”
BackgroundTowards zero knowledge argument for double discrete logarithm with constant cost
Krishnan, D., & Fu, X. (2024) · Theoretical Computer Science · doi:10.1016/j.tcs.2024.114799 · Cited by 1
Wang, Y. (2024) · Computers & Electrical Engineering · doi:10.1016/j.compeleceng.2024.109723 · Cited by 1
Batch Validation Scheme of Data Feature Requirement in Blockchain-Based Data Trading Platform
Zhang, J., Yu, Z., Chen, T., Fang, J., Liang, C., & Tang, H. (2024) · Communications in computer and information science · doi:10.1007/978-981-97-9412-6_42 · Cited by 1
Li, Y., Ren, X., Yin, C., Tang, C., Zhang, J., & Zheng, L. (2024) · Communications in computer and information science · doi:10.1007/978-981-97-9412-6_18 · Cited by 1
A Scheme for Privacy Protection of Medical Data Based on Blockchain
Gao, J., Guan, J., & Xu, J. (2024) · International Conference on Computer Science and Blockchain (CCSB) · doi:10.1109/ccsb63463.2024.10735482 · Cited by 1
“Bulletproofs (BP)[8], introduced by Benedikt Bünz is a non-interactive zero-knowledge proof protocol known for its very short proofs and lack of a trusted setup.”
BackgroundSuccinct Hash-Based Arbitrary-Range Proofs
Li, W., Zhang, Z., Guo, Y., Chow, S. S. M., & Wan, Z. (2024) · IEEE Transactions on Information Forensics and Security · doi:10.1109/tifs.2024.3497806 · Cited by 1
“In contrast, proofs (from other assumptions) specialized for ranges [11] and integer relations [15], [19] usually surpass generic ones.”
InfluentialV2I-Aided zk-SNARK for Travel Records Verification of Electric Vehicles
Ding, C., Ho, I. W., & Chau, C.-K. (2024) · IEEE International Symposium on Telecommunication Technologies · doi:10.1109/istt63363.2024.10750742 · Cited by 1
“To minimize the proof size, Bulletproofs protocol was proposed in [10].”
From One-Time to Two-Round Reusable Multi-signatures Without Nested Forking
Rotem, L., Segev, G., & Yogev, E. (2024) · Lecture notes in computer science · doi:10.1007/978-3-031-78020-2_13 · Cited by 1
Folding Schemes with Privacy Preserving Selective Verification
Boyar, J., & Erfurth, S. (2025) · IACR Communications in Cryptology · doi:10.62056/a0iv4fe-3 · Cited by 1
“As a first example, we consider Inner Product Relation of Committed Values [BCC + 16, RZ23], which are used in Bulletproofs [BBB + 18].”
MethodSD-SmartCert: A Blockchain-Based Certification System with Selective Disclosure
Modak, A., Jha, S., Patil, S. C., & Pise, R. G. (2024) · IEEE International Conference on Blockchain and Distributed Systems Security (ICBDS) · doi:10.1109/icbds61829.2024.10837562 · Cited by 1
Threshold Ring Signatures: From DualRing to the $$t+1$$ Rings
Yuen, T. H., & Pan, S. (2025) · Lecture notes in computer science · doi:10.1007/978-981-96-0954-3_14 · Cited by 1
Teng, D., Yao, Y., & Huang, C. (2025) · EURASIP Journal on Information Security · doi:10.1186/s13635-025-00192-9 · Cited by 1
DSKE: Digital Signatures with Key Extraction
Wang, Z., Alpos, O., Kavousi, A., Wong, H. W. H., Chau, S. Y., Le, D. V., & Cachin, C. (2025) · Lecture notes in computer science · doi:10.1007/978-3-031-88661-4_7 · Cited by 1
“Also, one may deploy polynomial commitment schemes that do not require trusted setup, such as the one constructed from Bulletproofs [BBB + 18].”
Dynamic Collateral Management in Decentralized Sealed-Bid Auctions
Abdullah, K., Iwamoto, M., & Watanabe, Y. (2024) · International Conference on Informatics, Multimedia, Cyber and Information System (ICIMCIS) · doi:10.1109/icimcis63449.2024.10957061 · Cited by 1
“The protocol revolves around cryptographic techniques like Pedersen commitments [23] and range proofs [24] to maintain the integrity of the bidding process while ensuring the confidentiality of each participant ’ s bid.”
Fast Payment System in Cryptocurrencies Through Off-Chain Transaction Aggregation
Zheng, L., Du, R., Chen, J., He, K., Jia, M., & Wang, Y. (2025) · IEEE Internet of Things Journal · doi:10.1109/jiot.2025.3578966 · Cited by 1
“Bulletproofs [26] is a range proof protocol that allows anyone to prove that multiple values are in the same range while hiding these values.”
Permissionless Verifiable Information Dispersal (Data Availability for Bitcoin Rollups)
Fisch, B., Lazzaretti, A., Liu, Z., & Yang, L. (2025) · IEEE Symposium on Security and Privacy (S&P) · doi:10.1109/sp61157.2025.00248 · Cited by 1
“Instead, honest participants broadcast their sampling results to others (i.e., a vote of yes or no on whether the sampling was successful) using some form of reliable broadcast (which ensures messages from honest participants are eventually received by all, and any received message is eventually…”
Zhang, M., Li, X., Zhang, J., Hóu, Z., Bai, G., & Feng, R. (2025) · International Conference on Computer Supported Cooperative Work in Design (CSCWD) · doi:10.1109/cscwd64889.2025.11033363 · Cited by 3
Efficient Byzantine Broadcast From Succinct Erasure Coding Proof System
Alhaddad, N., Duan, S., Varia, M., Wang, H., & Zhang, H. (2025) · IEEE Transactions on Information Forensics and Security · doi:10.1109/tifs.2025.3592564 · Cited by 1
“(2) ECP-2 is an alternate instantiation of Algorithm 1 based on polynomial commitment schemes with transparent setup and logarithmic-sized witnesses, such as Bul-letproofs [18] or the DARK polynomial commitment scheme [17].”
Protection of Electricity Trading Data Based on Blockchain Technology
Zhang, K., Liu, G., He, Z., Wu, C. H., Zhao, J., Yang, B., & Meng, S. (2025) · Journal of Electrical Engineering and Technology · doi:10.1007/s42835-025-02472-3 · Cited by 1
“This reduces the risk of sensitive information being compromised during data exchanges, thereby ensuring the integrity and security of the electricity information collection framework [22].”
A survey on threshold digital signature schemes
Peng, Y., Feng, Q., He, D., & Luo, M. (2025) · Frontiers of Computer Science · doi:10.1007/s11704-025-41297-1 · Cited by 1
Le, T., & Mutka, M. W. (2019) · EAI International Conference on Mobile and Ubiquitous Systems: Computing, Networking and Services (MobiQuitous) · doi:10.1145/3360774.3360794 · Cited by 1
“More recently, Bünz et al. propose Bulletproof [9], a much shorter and more efficient range proof.”
BackgroundMining for Privacy: How to Bootstrap a Snarky Blockchain
Kerber, T., Kiayias, A., & Kohlweiss, M. (2021) · Lecture notes in computer science · doi:10.1007/978-3-662-64322-8_24 · Cited by 1
“Agreatdeal of work has also gone into the design of non-interactive zero-knowledge which does not require structure in it’s references, such as DARK [10], STARKs [3], and Bulletproofs [9].”
BackgroundGemini-Chain: A Regulatable Digital Currency Model based on Blockchain
Zhang, J., Li, P., Xu, Z., Wang, A., Liu, J., & Wang, Z. (2020) · IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS) · doi:10.1109/infocomwkshps50562.2020.9163025 · Cited by 1
“[7] proposed an efficient noninteractive zero-knowledge proof protocol and used it in”
MethodBatching Anonymous and Non-Anonymous Membership Proofs for Blockchain Applications
Fu, S., Luo, G., & Gong, G. (2021) · IEEE World AI IoT Congress (AIIoT) · doi:10.1109/aiiot52608.2021.9454223 · Cited by 1
“A lot of research work on zkSNARK for general NP statements in the form of arithmetic circuit satisfiability has been done [5], [6], [8], [9], [13].”
BackgroundDistributed Computing in Blockchain Technology
Ramalingam, V., Mariappan, D. B., Premkumar, S., & Kumar, C. R. (2021) · EAI/Springer Innovations in Communication and Computing · doi:10.1007/978-3-030-70501-5_4 · Cited by 1
Artificial Intelligence in the Energy Industry
(2022) · MDPI eBooks · doi:10.3390/books978-3-0365-4606-3 · Cited by 1
Cryptographic obfuscation for smart contracts: Trustless bitcoin bridge and more
Suegami, S. (2022) · Blockchain Research and Applications · doi:10.1016/j.bcra.2022.100118 · Cited by 1
A Proposal for Enhancing the Security and Privacy of Digital Currencies
Takaragi, K., & Kubota, T. (2023) · Advances in public policy and administration (APPA) book series · doi:10.4018/978-1-6684-6429-8.ch010 · Cited by 1
KeyForge: Mitigating Email Breaches with Forward-Forgeable SignaturesPreprint
Specter, M. A., Park, S., & Green, M. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1904.06425 · Cited by 1
“More recent proving systems such as Bulletproofs [14] and zkSNARKs (e.”
BackgroundZKPs: Does This Make The Cut? Recent Advances and Success of Zero-Knowledge Security ProtocolsPreprint
Kassaras, S., & Maglaras, L. A. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2006.09990 · Cited by 1
Delegated RingCT: faster anonymous transactionsPreprint
Morais, R., Crocker, P. A., & de Sousa, S. M. (2020) · arXiv (Cornell University) · Cited by 1
Long-term privacy in electronic voting systems
Miranda, N. C. (2021) · doi:10.5821/dissertation-2117-348913 · Cited by 1
New and improved zero-knowledge argument systems for code-based cryptography
Tang, K. H. (2022) · doi:10.32657/10356/160385 · Cited by 1
Yu, C., Du, X., Wu, X., Yang, Q., Wang, W., & Wang, S. (2023) · SSRN Electronic Journal · doi:10.2139/ssrn.4668769 · Cited by 1
Harmonia: Securing Cross-Chain Applications Using Zero-Knowledge ProofsPreprint
Belchior, R., Dimov, D., Karadjov, Z., Pfannschmidt, J., Vasconcelos, A., & Correia, M. P. (2024) · doi:10.36227/techrxiv.170327806.66007684/v2 · Cited by 1
An Algebraic Framework for Universal and Updatable SNARKs
Ràfols, C., & Zapico, A. (2021) · Lecture notes in computer science · doi:10.1007/978-3-030-84242-0_27
SANS: Self-Sovereign Authentication for Network Slices
Salleras, X., & Daza, V. (2020) · Security and Communication Networks · doi:10.1155/2020/8823573
Proof of All: Verifiable Computation in a NutshellPreprint
Barbara, M. A. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1908.02327 · Cited by 1
“• Linear PCPs while this DLP-based (Discrete Logarithm Problem) field is not strictly universal, as proofs can only guarantee that given inputs lie within a specific range (e.g. boundary constraints), Bulletproofs [83] have often been compared to other universal systems because of their applicability to cryptocurrenci…”
BackgroundAcceptable Privacy for XAND Blockchain
Chawdhuri, D. R., Kramer, J., & Judge, S. (2019) · International Conference on Internet of Things: Systems, Management and Security (IOTSMS) · doi:10.1109/iotsms48152.2019.8939262
“• A proof of the total amount spent in the day being less than the maximum limit ((Cj , Dj),Cj , J, (C ′, D′), (A′, B′),C′, α2, γ) combined with a BulletProof and a Schnorr signature of an AP using an OR proof, so as to prove either one is a valid proof.”
MethodInfluentialAn Efficient Group Signature Based Digital Currency System
Tian, H., Luo, P., & Su, Y. (2020) · Communications in computer and information science · doi:10.1007/978-981-15-2767-8_34
Možnosti in izzivi uporabe protokola IPFS
Kamišalić, A., & Turkanović, M. (2019) · OTS Sodobne informacijske tehnologije in storitve (OTS) · doi:10.18690/978-961-286-282-4.9
Implementacija programskih rešitev s pomočjo inteligentnih asistentov
Gradišnik, M., Beranič, T., & Karakatič, S. (2019) · OTS Sodobne informacijske tehnologije in storitve (OTS) · doi:10.18690/978-961-286-282-4.16
Prihodnost Jave v luči zadnjih velikih sprememb
Krajnc, A., Petr, C., Skuhala, M., & Ramšak, G. (2019) · OTS Sodobne informacijske tehnologije in storitve (OTS) · doi:10.18690/978-961-286-282-4.24
CoinMagic: A Differential Privacy Framework for Ring Signature SchemesPreprint
Ni, W., Wu, H., Cheng, P., Chen, L., Lin, X., Chen, L., Lai, X., & Zhang, X. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2003.06826
“In [14], the researchers put forward the newest RingCT protocol based on Bullet-proof [38], which decreases the RS’s size from O ( n ) to O (log n ) .”
Legally Binding Anonymous Multiparty Commitments on a Blockchain
Chawdhuri, D. R. (2020) · Crypto Valley Conference on Blockchain Technology (CVCBT) · doi:10.1109/cvcbt50464.2020.00009
Halo 0.9: A Halo Protocol with Fully-Succinctness.Preprint
Wang, L. (2020) · IACR Cryptology ePrint Archive
Supersonic SNARKs and Applications
Anh, D. H., Nguyen, H. V. Q., & Ngo, H. D. (2020) · NAFOSTED Conference on Information and Computer Science (NICS) · doi:10.1109/nics51282.2020.9335838
“Transparent proof systems usually comes with compromises on verifier performance and proof size, two examples are zk-STARKs [1] and Bulletproof [4].”
Scaling Distributed Ledgers and Privacy-Preserving ApplicationsPreprint
Sonnino, A. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2102.12273
Jeong, S., & Ahn, B. (2021) · Computer Science and Information Systems · doi:10.2298/csis200322006j
“Our goal is to minimize the cost of these pairing assessments [17].”
BackgroundA DLT-based Smart Contract Architecture for Atomic and Scalable TradingPreprint
Kalbantner, J., Markantonakis, K., Hurley-Smith, D., Shepherd, C., & Semal, B. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2105.02937
“zk-SNARK (zero-knowledge succinct non-interactive argument of knowledge) [50], ring signatures [51], [52], [53] and bulletproofs (non-interactive zero-knowledge proofs) [54].”
Lelantus Spark: Secure and Flexible Private Transactions
Jivanyan, A., & Feickert, A. J. (2023) · Lecture notes in computer science · doi:10.1007/978-3-031-32415-4_28
“In practice, an efficient instantiation like Bulletproofs [8] or Bulletproofs+ [9] may be used to satisfy this requirement.”
MethodKey Exposure Resistant Group Key Agreement Protocol
Zhou, T., Shen, J., Ji, S., Ren, Y., & Zhang, M. (2021) · Lecture notes in computer science · doi:10.1007/978-3-030-90402-9_5
(∈, δ)-Indistinguishable Mixing for Cryptocurrencies
Liang, M., Karantaidou, I., Baldimtsi, F., Gordon, S., & Varia, M. (2021) · Proceedings on Privacy Enhancing Technologies · doi:10.2478/popets-2022-0004
Kim, S. (2022) · Electronics · doi:10.3390/electronics11010131
“[23] studied additive polynomial commitment schemes, where commitments form an additive group [6,10,14,19,22].”
BackgroundMethodFerproof: A Constant Cost Range Proof Suitable for Floating-Point Numbers
Li, Y., Zhou, K., Xu, L., Wang, M., & Liu, N. (2022) · Lecture notes in computer science · doi:10.1007/978-3-030-95391-1_41
Scriptable and composable SNARKs in the trusted hardware model1
Zhou, Z., Zhang, B., Chen, Y., Li, J., Zhou, Y., Lu, Y., Ren, K., Thai, P. D., & Zhou, H.-S. (2022) · Journal of Computer Security · doi:10.3233/jcs-210167
“The second approach is based on the hardness of the DLP, originally proposed by Groth [29] and then implemented in [9,12].”
Hydra: Pipelineable Interactive Arguments of Knowledge for Verifiable Neural Networks
Zhang, W., & Xia, Y. (2021) · IEEE International Conference on Trust, Privacy and Security in Intelligent Systems and Applications (TPS-ISA) · doi:10.1109/tpsisa52974.2021.00001
Overview of Block Chain Privacy Protection Methods
Mu, K., Ma, L., Hou, Y., Lu, Z., Yang, H., & Huang, A. (2022) · International Conference on Measuring Technology and Mechatronics Automation (ICMTMA) · doi:10.1109/icmtma54903.2022.00191
Metadata
- Authors
- Bunz, B., Bootle, J., Boneh, D., Poelstra, A., Wuille, P., & Maxwell, G.
- Affiliations
- University College London (GB)
- Stanford University (US)
- Venue
- 2018 IEEE Symposium on Security and Privacy (SP)
- Publisher
- IEEE
- Published
- 2018-05
- Pages
- 315-334
- ISSN
- 1081-6011 (ISSN-L)
- OpenAlex
- W2883748642
- Cited by
- 1,154
- References
- 63
- Citations by year
- 2026: 148 · 2025: 184 · 2024: 199 · 2023: 175 · 2022: 147 · 2021: 128 · 2020: 104 · 2019: 60 · 2018: 9
- FWCI
- 30.72
- Type
- conference paper
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- IEEE Symposium on Security and Privacy (S&P) (publishedVersion)
- Funders
- National Science Foundation
- Simons Foundation
- Defense Advanced Research Projects Agency
- Office of Naval Research
- Topics
- Physical Sciences › Computer Science › Computer Networks and Communications › Cryptography and Data Security
- Physical Sciences › Computer Science › Information Systems › Blockchain Technology Applications and Security
- Physical Sciences › Computer Science › Computer Networks and Communications › Advanced Authentication Protocols Security
- Keywords
- Bulletproofs, non-interactive zero-knowledge proofs, range proofs, confidential transactions, Bitcoin, multi-party computation, arithmetic circuits, discrete logarithm assumption, blockchain privacy
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