An Extended Survey Concerning the Vector Commitments
Source abstract
Abstract
Commitment schemes represent foundational cryptographic primitives enabling secure verification protocols across diverse applications, from blockchain systems to zero-knowledge proofs. This paper presents a systematic survey of vector, polynomial, and functional commitment schemes, analyzing their evolution from classical constructions to post-quantum secure alternatives. We examine the strengths and limitations of RSA-based, Diffie–Hellman, and lattice-based approaches, highlighting the critical shift toward quantum-resistant designs necessitated by emerging computational threats. The survey reveals that while lattice-based schemes (particularly those using the Short Integer Solution problem) offer promising security guarantees, they face practical challenges in proof size and verification efficiency. Functional commitments emerge as a powerful generalization, though their adoption is constrained by computational overhead and setup requirements. Key findings identify persistent gaps in adaptive security, composability, and real-world deployment, while proposed solutions emphasize optimization techniques and hybrid approaches. By synthesizing over 90 research works, this paper provides both a comprehensive reference for cryptographic researchers and a roadmap for future developments in commitment schemes, particularly in addressing the urgent demands of post-quantum cryptography and decentralized systems.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Cryptography and Data Security · Complexity and Algorithms in Graphs · Privacy-Preserving Technologies in Data
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
Affiliations
Department of Mathemathics and Computer Science, Faculty of Mathematics and Computer Science, Transilvania University of Brasov, 500036 Brașov, Romania
School of Business, Technology and Education, Ilia State University, Tbilisi 0179, Georgia
School of Technology, Caucasus University, Tbilisi 0162, Georgia
Funding
NATO Science for Peace and Security (SPS)
References (98)
Show source-provided references
1. Catalano, D., and Fiore, D. (2013). Vector commitments and their applications. Public-Key Cryptography–PKC 2013: 16th International Conference on Practice and Theory in Public-Key Cryptography, Nara, Japan, February 26–March 1, 2013; Proceedings 16, Springer.
2. Underwater Color Disparities: Cues for Enhancing Underwater Images Toward Natural Color Consistencies10.1109/TCSVT.2023.3289566 ↗
3. A method for obtaining digital signatures and public-key cryptosystems10.1145/359340.359342 ↗
4. Systematic and Critical Review of RSA Based Public Key Cryptographic Schemes: Past and Present Status10.1109/ACCESS.2021.3129224 ↗
5. Paixao, C.A.M., and Filho, D.L.G. (2005). An efficient variant of the RSA cryptosystem. Simpósio Brasileiro em Segurança da Informação e de Sistemas Computacionais (SBSeg), Springer.10.5753/sbseg.2005.21521 ↗
6. Fast variants of rsa
7. Karakra, A., and Alsadeh, A. (2016, January 13–15). A-RSA: Augmented RSA. Proceedings of the 2016 SAI Computing Conference (SAI), London, UK.10.1109/SAI.2016.7556103 ↗
8. Rabin, M.O. (1979). Digitalized Signatures and Public-Key Functions as Intractable as Factorization, Massachusetts Institute of Technology. MIT Technical Report (1979).
9. A Method for the Construction of Minimum-Redundancy Codes10.1109/JRPROC.1952.273898 ↗
10. Jintcharadze, E., and Iavich, M. (2020, January 4–7). Hybrid implementation of TWOFISH, AES, ELGAMAL and RSA cryptosystems. Proceedings of the 2020 IEEE East-West Design and Test Symposium (EWDTS), Varna, Bulgaria.10.1109/EWDTS50664.2020.9224901 ↗
11. A Hybrid Encryption Algorithm Based on RSA Small-e and Efficient-RSA for Cloud Computing Environments10.7763/JACN.2013.V1.47 ↗
12. Bansal, V.P., and Singh, S. (2015, January 21–22). A hybrid data encryption technique using RSA and blowfish for cloud computing on FPGAS. Proceedings of the 2015 2nd International Conference on Recent Advances in Engineering and Computational Sciences (RAECS), Chandigarh, India.10.1109/RAECS.2015.7453367 ↗
13. The blowfish encryption algorithm
14. El Makkaoui, K., Ezzati, A., and Beni-Hssane, A. (2017). Cloud-RSA: An enhanced homomorphic encryption scheme. Europe and MENA Cooperation Advances in Information and Communication Technologies, Springer.10.1007/978-3-319-46568-5_48 ↗
15. Fast cloud-RSA scheme for promoting data confidentiality in the cloud computing10.1016/j.procs.2017.08.282 ↗
16. A new approach for image encryption in the modified RSA cryptosystem using MATLAB
17. Estimation of Modified RSA Cryptosystem with Hyper Image Encryption Algorithm10.17485/ijst/2017/v10i7/111000 ↗
18. Development of modified RSA algorithm using fixed MERSENNE prime numbers for medical ultra-sound imaging instrumentation10.1080/24699322.2019.1649070 ↗
19. A visually secure asymmetric image encryption scheme based on RSA algorithm and hyperchaotic map10.1088/1402-4896/ab52bc ↗
20. Image Encryption Scheme Based on a Generalized Arnold Map and RSA Algorithm
21. A novel multi-layer color image encryption based on RSA cryptosystem, RP2DFrHT and generalized 2D Arnold map10.1007/s11042-023-14829-9 ↗
22. Soni, G.K., Arora, H., and Jain, B. (2020). A novel image encryption technique using ARNOLD transform and asymmetric RSA algorithm. International Conference on Artificial Intelligence: Advances and Applications 2019: Proceedings of ICAIAA 2019, Springer.10.1007/978-981-15-1059-5_10 ↗
23. Preferred reporting items for systematic reviews and meta-analyses: The prisma statement10.1016/j.ijsu.2010.02.007 ↗
24. Si, H., Cai, Y., and Cheng, Z. (2010, January 6–7). An improved RSA signature algorithm based on complex numeric operation function. Proceedings of the 2010 International Conference on Challenges in Environmental Science and Computer Engineering, Wuhan, China.10.1109/CESCE.2010.257 ↗
25. Aufa, F.J., and Affandi, A. (2018, January 7–8). Security system analysis in combination method: RSA encryption and digital signature algorithm. Proceedings of the 2018 4th International Conference on Science and Technology (ICST), Yogyakarta, Indonesia.10.1109/ICSTC.2018.8528584 ↗
26. Frunza, M., and Scripcariu, L. (2007, January 13–14). Improved RSA encryption algorithm for increased security of wireless networks. Proceedings of the 2007 International Symposium on Signals, Circuits and Systems, Iasi, Romania.10.1109/ISSCS.2007.4292737 ↗
27. Fiat, A. (1990). Batch RSA. Advances in Cryptology—CRYPTO’89 Proceedings 9, Springer.10.1007/0-387-34805-0_17 ↗
28. Li, Y., Liu, Q., and Li, T. (2010, January 9–11). Design and implementation of two improved batch RSA algorithms. Proceedings of the 2010 3rd International Conference on Computer Science and Information Technology, Chengdu, China.
29. Collision resistance from multi-collision resistance10.1007/s00145-024-09495-5 ↗
30. Peikert, C., Pepin, Z., and Sharp, C. (2025, June 18). Vector and Functional Commitments from Lattices. Cryptology ePrint Archive, Article 2021/1254. Available online: https://eprint.iacr.org/2021/1254.
31. Pedersen, T.P. (1991). Non-interactive and information-theoretic secure verifiable secret sharing. Annual International Cryptology Conference, Springer.
32. A public key cryptosystem and a signature scheme based on discrete logarithms10.1109/TIT.1985.1057074 ↗
33. Bao, F., Deng, R.H., and Zhu, H. (2003). Variations of diffie-hellman problem. Proceedings of the International Conference on Information and Communications Security, Springer.10.1007/978-3-540-39927-8_28 ↗
34. The twin diffie–hellman problem and applications10.1007/s00145-009-9041-6 ↗
35. Canetti, R., Gennaro, R., Goldfeder, S., Makriyannis, N., and Peled, U. (2020, January 9–13). Uc non-interactive, proactive, threshold ECDSA with identifiable aborts. Proceedings of the 2020 ACM SIGSAC Conference on Computer and Communications Security, Virtual Event.10.1145/3372297.3423367 ↗
36. Metere, R., and Dong, C. (2017). Automated cryptographic analysis of the PEDERSEN commitment scheme. arXiv.10.1007/978-3-319-65127-9_22 ↗
37. Barthe, G., Dupressoir, F., Gr’egoire, B., Kunz, C., Schmidt, B., and Strub, P.-Y. (2012, January 3–8). Easycrypt: A tutorial. Proceedings of the International School on Foundations of Security Analysis and Design, Bertinoro, Italy.
38. SBRAC: Blockchain-based sealed-bid auction with bidding price privacy and public verifiability
39. Wang, H., and Liao, J. (2022, January 17–19). Blockchain privacy protection algorithm based on pedersen commitment and zero-knowledge proof. Proceedings of the 2021 4th International Conference on Blockchain Technology and Applications, ICBTA ’21, Xi’an, China.10.1145/3510487.3510488 ↗
40. Liu, F., Yang, J., Kong, D., and Qi, J. (2020, January 28–31). A secure multi-party computation protocol combines pederson commitment with schnorr signature for blockchain. Proceedings of the 2020 IEEE 20th International Conference on Communication Technology (ICCT), Nanning, China.
41. Time controlled expressive predicate query with accountable anonymity10.1109/TSC.2022.3162285 ↗
42. Shor, P.W. (1994, January 20–22). Algorithms for Quantum Computation: Discrete Logarithms and Factoring. Proceedings of the 35th Annual Symposium on Foundation of Computer Science, Washington, DC, USA.
43. A decade of lattice cryptography10.1561/0400000074 ↗
44. Li, Y., Ng, K.S., and Purcell, M. (2022). A tutorial introduction to lattice-based cryptography and homomorphic encryption. arXiv.
45. Ajtai, M. (1996, January 22–24). Generating hard instances of lattice problems. Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Philadelphia, PA, USA.10.1145/237814.237838 ↗
46. Berman, I., Degwekar, A., Rothblum, R.D., and Vasudevan, P.N. (2018). Multi-collision resistant hash functions and their applications. Advances in Cryptology–EUROCRYPT 2018: 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tel Aviv, Israel, April 29–May 3, 2018; Proceedings, Part II 37, Springer.
47. White-box vs. black-box complexity of search problems: Ramsey and graph property testing10.1145/3341106 ↗
48. Bitansky, N., Kalai, Y.T., and Paneth, O. (2018, January 25–29). Multi-collision resistance: A paradigm for keyless hash functions. Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, Los Angeles, CA, USA.10.1145/3188745.3188870 ↗
49. Komargodski, I., Naor, M., and Yogev, E. (2018). Collision resistant hashing for paranoids: Dealing with multiple collisions. Advances in Cryptology—EUROCRYPT 2018: 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Tel Aviv, Israel, April 29–May 3, 2018; Proceedings, Part II 37, Springer.10.1007/978-3-319-78375-8_6 ↗
50. Komargodski, I., and Yogev, E. (2018, January 19–23). On distributional collision resistant hashing. Proceedings of the Annual International Cryptology Conference, Santa Barbara, CA, USA.10.1007/978-3-319-96881-0_11 ↗
51. Papamanthou, C., Shi, E., Tamassia, R., and Yi, K. (2013). Streaming authenticated data structures. Advances in Cryptology—EU-ROCRYPT 2013, Springer. Volume 7881 of Lecture Notes in Computer Science.10.1007/978-3-642-38348-9_22 ↗
52. Boneh, D., Bünz, B., and Fisch, B. (2019). Batching techniques for accumulators with applications to IOPs and stateless blockchains. Advances in Cryptology–CRYPTO 2019: 39th Annual International Cryptology Conference, Santa Barbara, CA, USA, 18–22 August 2019; Proceedings, Part I 39, Springer.10.1007/978-3-030-26948-7_20 ↗
53. Tas, E.N., and Boneh, D. (2023). Vector commitments with efficient updates. arXiv.
54. A (zero-knowledge) vector commitment with sum binding and its applications10.1093/comjnl/bxz115 ↗
55. Publicly verifiable databases with efficient insertion/deletion operations10.1016/j.jcss.2016.07.005 ↗
56. Kate, A., Zaverucha, G.M., and Goldberg, I. (2010, January 5–9). Constant-Size Commitments to Polynomials and Their Applications. Proceedings of the ASIACRYPT 2010: 16th International Conference on the Theory and Application of Cryptology and Information Security, Singapore.
57. Belling, A., Soleimanian, A., and Ursu, B. (2025, June 18). Vortex: A List Polynomial Commitment and Its Application to Arguments of Knowledge. Cryptology ePrint Archive, 2024 Article 2024/185. Available online: https://eprint.iacr.org/2025/185.
58. Golovnev, A., Lee, J., Setty, S., Thaler, J., and Wahby, R.S. (2023). Brakedown: Linear-time and field-agnostic snarks for R1CS. 43rd Annual International Cryptology Conference, CRYPTO 2023, Santa Barbara, CA, USA, 20–24 August 2023, Proceedings, Part II, Springer.10.1007/978-3-031-38545-2_7 ↗
59. Kattis, A.A., Panarin, K., and Vlasov, A. (2022, January 7–11). Redshift: Transparent snarks from list polynomial commitments. Proceedings of the 2022 ACM SIGSAC Conference on Computer and Communications Security, Los Angeles, CA, USA.10.1145/3548606.3560657 ↗
60. Belohorec, J., Dvor’ak, P., Hoffmann, C., Hub’acek, P., Maskov’a, K., and Pastyr’ık, M. (2025, May 08). On Extractability of the kzg Family of Polynomial Commitment Schemes. Cryptology ePrint Archive, 2025, Article 2025/514. Available online: https://eprint.iacr.org/2025/514.
61. Seres, I.A., and Burcsi, P. (2025, May 07). Behemoth: Transparent Polynomial Commitment Scheme with Constant Opening Proof Size and Verifier Time. Cryptology ePrint Archive, 2023, Article 2023/670. Available online: https://eprint.iacr.org/2023/670.
62. Nikolaenko, V., Ragsdale, S., Bonneau, J., and Boneh, D. (2024, January 5–8). Powers-of-tau to the people: Decentralizing setup ceremonies. Proceedings of the International Conference on Applied Cryptography and Network Security, Abu Dhabi, United Arab Emirates.10.1007/978-3-031-54776-8_5 ↗
63. Avery, C., and Sheek, J. (2025, May 27). Storage Auditing Using Merkle Trees and KZG Commitments. Available online: https://www.orchid.com/storage-auditing-latest.pdf.
64. Lee, J. (2021). Dory: Efficient, transparent arguments for generalised inner products and polynomial commitments. Theory of Cryptography, Springer.10.1007/978-3-030-90453-1_1 ↗
65. Kuszmaul, J. (2025, May 17). Verkle Trees. Verkle Trees 2018, Kuszmaul, J. Verkle Trees. Available online: https://math.mit.edu/research/highschool/primes/materials/2018/Kuszmaul.pdf.
66. Blum, M., Feldman, P., and Micali, S. (2019). Non-interactive zero- knowledge and its applications. Providing Sound Foundations for Cryptography: On the Work of Shafi Goldwasser and Silvio Micali, Association for Computing Machinery.
67. Bitansky, N., Canetti, R., Chiesa, A., and Tromer, E. (2012, January 8–10). From extractable collision resistance to succinct non-interactive arguments of knowledge, and back again. Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, Cambridge, MA, USA.10.1145/2090236.2090263 ↗
68. Sasson, E.B., Chiesa, A., Garman, C., Green, M., Miers, I., Tromer, E., and Virza, M. (2014, January 18–21). Zerocash: Decentralized anonymous payments from bitcoin. Proceedings of the 2014 IEEE Symposium on Security and Privacy, Berkeley, CA, USA.
69. Hopwood, D., Bowe, S., Hornby, T., and Wilcox, N. (2025, April 01). Zcash Protocol Specification, Version 2024.1.1. Available online: https://zips.z.cash/protocol/protocol.pdf.
70. Ben-Sasson, E., Chiesa, A., Riabzev, M., Spooner, N., Virza, M., and Ward, N.P. (2019). Aurora: Transparent succinct arguments for r1cs. Advances in Cryptology–EUROCRYPT 2019: 38th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Darmstadt, Germany, May 19–23, 2019; Proceedings, Part I 38, Springer.10.1007/978-3-030-17653-2_4 ↗
71. estark: Extending starks with arguments10.1007/s10623-024-01457-z ↗
72. Ashur, T., and Dhooghe, S. (2025, May 19). Marvellous: A Stark-Friendly Family of Cryptographic Primitives. Cryptology ePrint Archive, 2018. Article 2018/1098. Available online: https://eprint.iacr.org/2018/1098.
73. Ben-Sasson, E., Bentov, I., Horesh, Y., and Riabzev, M. (2018, January 9–13). Fast reed-solomon interactive oracle proofs of proximity. Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018), Prague, Czech Republic.
74. Ames, S., Hazay, C., Ishai, Y., and Venkita-subramaniam, M. (November, January 30). Ligero: Lightweight sublinear arguments without a trusted setup. Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security, Dallas, TX, USA.10.1145/3133956.3134104 ↗
75. Xie, T., Zhang, Y., and Song, D. (2022, January 15–18). Orion: Zero knowledge proof with linear prover time. Proceedings of the Annual International Cryptology Conference, Santa Barbara, CA, USA.10.1007/978-3-031-15985-5_11 ↗
76. Bootle, J., Cerulli, A., Ghadafi, E., Groth, J., Hajiabadi, M., and Jakobsen, S.K. (2017, January 3–7). Linear-time zero-knowledge proofs for arithmetic circuit satisfiability. Proceedings of the International Conference on the Theory and Application of Cryptology and Information Security, Hong Kong, China.10.1007/978-3-319-70700-6_12 ↗
77. Bootle, J., Chiesa, A., and Groth, J. (2020, January 16–19). Linear-time arguments with sublinear verification from tensor codes. Proceedings of the Theory of Cryptography—18th International Conference, TCC 2020, Durham, NC, USA.10.1007/978-3-030-64378-2_2 ↗
78. Bootle, J., Chiesa, A., and Liu, S. (2022). Zero-knowledge IOPs with linear-time prover and polylogarithmic-time verifier. Annual International Conference on the Theory and Applications of Cryptographic Techniques, Springer.10.1007/978-3-031-07085-3_10 ↗
79. den Hollander, T., and Slamanig, D. (2025, May 12). A Crack in the Firmament: Restoring Soundness of the Orion Proof System and More. Cryptology ePrint Archive, 2024, Article 2024/1164. Available online: https://eprint.iacr.org/2024/1164.
80. Diamond, B.E., and Posen, J. (2025, June 11). Proximity Testing with Logarithmic Randomness. Cryptology ePrint Archive, 2023, Article 2023/630. Available online: https://eprint.iacr.org/2023/630.
81. Setty, S. Spartan: Efficient and general-purpose ZKSNARKs without trusted setup. Proceedings of the 40th Annual International Cryptology Conference, CRYPTO 2020, Santa Barbara, CA, USA, 17–21 August 2020; Proceedings, Part III.10.1007/978-3-030-56877-1_25 ↗
82. Fu, S., and Gong, G. (2022, January 11–14). Polaris: Transparent succinct zero-knowledge arguments for r1cs with efficient verifier. Proceedings of the Privacy Enhancing Technologies, Virtual.10.2478/popets-2022-0027 ↗
83. Kilian, J. (1992, January 4–6). A note on efficient zero-knowledge proofs and arguments. Proceedings of the Twenty-Fourth Annual ACM Symposium on Theory of Computing, Victoria, BC, Canada.10.1145/129712.129782 ↗
84. Albrecht, M.R., Cini, V., Lai, R.W.F., Malavolta, G., and Thyagarajan, S.A. (2025, June 11). Lattice-Based SNARKs: Publicly Verifiable, Preprocessing, and Recursively Composable. Cryptology ePrint Archive 2022, Article 2022/941. Available online: https://eprint.iacr.org/2022/941.
85. de Castro, L., and Peikert, C. (2023). Functional commitments for all functions, with transparent setup and from sis. Annual International Conference on the Theory and Applications of Cryptographic Techniques, Springer.10.1007/978-3-031-30620-4_10 ↗
86. Canetti, R., and Fischlin, M. (2001). Universally composable commitments. Advances in Cryptology—CRYPTO 2001: 21st Annual International Cryptology Conference, Santa Barbara, California, USA, 19–23 August 2001 Proceedings 21, Springer.10.1007/3-540-44647-8_2 ↗
87. Srinivasan, S., Karantaidou, I., Baldimtsi, F., and Papamanthou, C. (2022, January 7–11). Batching, aggregation, and zero-knowledge proofs in bilinear accumulators. Proceedings of the 2022 ACM SIGSAC Conference on Computer and Communications Security, Los Angeles, CA, USA.10.1145/3548606.3560676 ↗
88. Du, W., and Atallah, M.J. (2001, January 10–13). Secure multi-party computation problems and their applications: A review and open problems. Proceedings of the 2001 Workshop on New Security Paradigms, Cloudcroft, Mexico.10.1145/508171.508174 ↗
89. Wee, H., and Wu, D.J. (2024, January 26–30). Succinct functional commitments for circuits from k-lin. Proceedings of the 43rd Annual International Conference on the Theory and Applications of Cryptographic Techniques, Zurich, Switzerland.10.1007/978-3-031-58723-8_10 ↗
90. Wee, H., and Wu, D.J. (2023, January 23–27). Succinct vector, polynomial, and functional commitments from lattices. Proceedings of the 42nd Annual International Conference on the Theory and Applications of Cryptographic Techniques, Lyon, France.10.1007/978-3-031-30620-4_13 ↗
91. Peikert, C., Pepin, Z., and Sharp, C. (2021). Vector and functional commitments from lattices. Theory of Cryptography: 19th International Conference, TCC 2021, Raleigh, NC, USA, November 8–11, 2021, Proceedings, Part III 19, Springer.
92. Boneh, D., Nguyen, W., and Ozdemir, A. (2025, April 01). Efficient Functional Commitments: How to Commit to a Private Function. Cryptology ePrint Archive, 2021, Article 2021/1342. Available online: https://eprint.iacr.org/2021/1342.
93. Wee, H., and Wu, D.J. (2023, January 4–8). Lattice-based functional commitments: Fast verification and cryptanalysis. Proceedings of the 29th International Conference on the Theory and Application of Cryptology and Information Security, Guangzhou, China.10.1007/978-981-99-8733-7_7 ↗
94. Lin, H., Pass, R., and Venkitasubramaniam, M. (2008). Concurrent non-malleable commitments from any one-way function. Theory of Cryptography: Fifth Theory of Cryptography Conference, TCC 2008, New York, USA, 19–21 March 2008; Proceedings 5, Springer.
95. Namboothiry, B. (2025, June 02). Revealable Functional Commitments: How to Partially Reveal a Secret Function. Cryptology ePrint Archive, 2023, Article 2023/1250. Available online: https://eprint.iacr.org/2023/1250.
96. Semantically enhanced selective image encryption scheme with parallel computing10.1016/j.eswa.2025.127404 ↗
97. A parallel image encryption algorithm using intra bitplane scrambling10.1016/j.matcom.2022.07.029 ↗
98. Batch image encryption using cross image permutation and diffusion