Communications in Cryptology IACR CiC

Synchronous Distributed Key Generation without Broadcasts

Authors

Nibesh Shrestha, Adithya Bhat, Aniket Kate, Kartik Nayak
Nibesh Shrestha
Supra Research, USA
nibeshrestha2 at gmail dot com
Adithya Bhat
Visa Research, USA
haxolotl dot research at gmail dot com
Aniket Kate
Supra Research, USA
Purdue University, USA
aniket at purdue dot edu
Kartik Nayak
Duke University, USA
kartik at cs dot duke dot edu

Abstract

Distributed key generation (DKG) is a key building block in developing many efficient threshold cryptosystems. This work initiates the study of communication complexity and round complexity of DKG protocols over a point-to-point (bounded) synchronous network. Our key result is the first synchronous DKG protocol for discrete log-based cryptosystems with $O(\kappa n^3)$ communication complexity ($\kappa$ denotes a security parameter) that tolerates any $t < n/2$ Byzantine faults among $n$ parties. We present two variants of the protocol: (i) a protocol with worst-case $O(\kappa n^3)$ communication and $O(t)$ rounds, and (ii) a protocol with expected $O(\kappa n^3)$ communication and expected constant rounds. In the process of achieving our results, we design (1) a novel weak gradecast protocol with a communication complexity of $O(\kappa n^2)$ for linear-sized inputs and constant rounds, (2) a protocol called “recoverable-set-of-shares” for ensuring recovery of shared secrets, (3) an oblivious leader election protocol with $O(\kappa n^3)$ communication and constant rounds, and (4) a multi-valued validated Byzantine agreement (MVBA) protocol with $O(\kappa n^3)$ communication complexity for linear-sized inputs and expected constant rounds. Each of these primitives is of independent interest.

References

[AAPP22]
Ittai Abraham, Gilad Asharov, Shravani Patil, and Arpita Patra. Asymptotically Free Broadcast in Constant Expected Time via Packed VSS. In Eike Kiltz and Vinod Vaikuntanathan, editors, TCC 2022: 20th Theory of Cryptography Conference, Part I, volume 13747 of Lecture Notes in Computer Science, pages 384–414, Chicago, IL, USA. 2022. Springer, Heidelberg, Germany. DOI: 10.1007/978-3-031-22318-1_14
[ADD+19]
Ittai Abraham, Srinivas Devadas, Danny Dolev, Kartik Nayak, and Ling Ren. Synchronous Byzantine Agreement with Expected $O(1)$ Rounds, Expected $O(n^2)$ Communication, and Optimal Resilience. In Ian Goldberg and Tyler Moore, editors, FC 2019: 23rd International Conference on Financial Cryptography and Data Security, volume 11598 of Lecture Notes in Computer Science, pages 320–334, Frigate Bay, St. Kitts and Nevis. 2019. Springer, Heidelberg, Germany. DOI: 10.1007/978-3-030-32101-7_20
[AJM+21]
Ittai Abraham, Philipp Jovanovic, Mary Maller, Sarah Meiklejohn, Gilad Stern, and Alin Tomescu. Reaching Consensus for Asynchronous Distributed Key Generation. In Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing, pages 363–373, New York, NY, USA. 2021. Association for Computing Machinery. DOI: 10.1145/3465084.3467914
[AJM+23]
Ittai Abraham, Philipp Jovanovic, Mary Maller, Sarah Meiklejohn, and Gilad Stern. Bingo: Adaptivity and Asynchrony in Verifiable Secret Sharing and Distributed Key Generation. In Advances in Cryptology – CRYPTO 2023: 43rd Annual International Cryptology Conference, CRYPTO 2023, Santa Barbara, CA, USA, August 20–24, 2023, Proceedings, Part I, pages 39–70, Berlin, Heidelberg. 2023. Springer-Verlag. DOI: 10.1007/978-3-031-38557-5_2
[AMS19]
Ittai Abraham, Dahlia Malkhi, and Alexander Spiegelman. Asymptotically Optimal Validated Asynchronous Byzantine Agreement. In Peter Robinson and Faith Ellen, editors, 38th ACM Symposium Annual on Principles of Distributed Computing, pages 337–346, Toronto, ON, Canada. 2019. Association for Computing Machinery. DOI: 10.1145/3293611.3331612
[BB08]
Dan Boneh and Xavier Boyen. Short Signatures Without Random Oracles and the SDH Assumption in Bilinear Groups. Journal of Cryptology, 21(2):149–177, April 2008. DOI: 10.1007/s00145-007-9005-7
[BCLZL23]
Renas Bacho, Daniel Collins, Chen-Da Liu-Zhang, and Julian Loss. Network-Agnostic Security Comes (Almost) for Free in DKG and MPC. In Advances in Cryptology – CRYPTO 2023: 43rd Annual International Cryptology Conference, CRYPTO 2023, Santa Barbara, CA, USA, August 20–24, 2023, Proceedings, Part I, pages 71–106, Berlin, Heidelberg. 2023. Springer-Verlag. DOI: 10.1007/978-3-031-38557-5_3
[BKP11]
Michael Backes, Aniket Kate, and Arpita Patra. Computational Verifiable Secret Sharing Revisited. In Dong Hoon Lee and Xiaoyun Wang, editors, Advances in Cryptology – ASIACRYPT 2011, volume 7073 of Lecture Notes in Computer Science, pages 590–609, Seoul, South Korea. 2011. Springer, Heidelberg, Germany. DOI: 10.1007/978-3-642-25385-0_32
[BL22]
Renas Bacho and Julian Loss. On the Adaptive Security of the Threshold BLS Signature Scheme. In Proceedings of the 2022 ACM SIGSAC Conference on Computer and Communications Security, pages 193–207, New York, NY, USA. 2022. Association for Computing Machinery. DOI: 10.1145/3548606.3560656
[BLL+23]
Renas Bacho, Christoph Lenzen, Julian Loss, Simon Ochsenreither, and Dimitrios Papachristoudis. GRandLine: Adaptively Secure DKG and Randomness Beacon with (Almost) Quadratic Communication Complexity. IACR Cryptol. ePrint Arch., 2023. https://eprint.iacr.org/2023/1887
[BOEY03]
Michael Ben-Or and Ran El-Yaniv. Resilient-optimal interactive consistency in constant time. Distributed Computing, 16(4):249–262, 2003. DOI: 10.1007/s00446-002-0083-3
[Bol03]
Alexandra Boldyreva. Threshold Signatures, Multisignatures and Blind Signatures Based on the Gap-Diffie-Hellman-Group Signature Scheme. In Yvo Desmedt, editor, PKC 2003: 6th International Workshop on Theory and Practice in Public Key Cryptography, volume 2567 of Lecture Notes in Computer Science, pages 31–46, Miami, FL, USA. 2003. Springer, Heidelberg, Germany. DOI: 10.1007/3-540-36288-6_3
[BP97]
Niko Bari and Birgit Pfitzmann. Collision-Free Accumulators and Fail-Stop Signature Schemes Without Trees. In Walter Fumy, editor, Advances in Cryptology – EUROCRYPT'97, volume 1233 of Lecture Notes in Computer Science, pages 480–494, Konstanz, Germany. 1997. Springer, Heidelberg, Germany. DOI: 10.1007/3-540-69053-0_33
[BSL+21]
Adithya Bhat, Nibesh Shrestha, Zhongtang Luo, Aniket Kate, and Kartik Nayak. RandPiper - Reconfiguration-Friendly Random Beacons with Quadratic Communication. In Giovanni Vigna and Elaine Shi, editors, ACM CCS 2021: 28th Conference on Computer and Communications Security, pages 3502–3524, Virtual Event, Republic of Korea. 2021. ACM Press. DOI: 10.1145/3460120.3484574
[CD17]
Ignacio Cascudo and Bernardo David. SCRAPE: Scalable Randomness Attested by Public Entities. In Dieter Gollmann, Atsuko Miyaji, and Hiroaki Kikuchi, editors, ACNS 17: 15th International Conference on Applied Cryptography and Network Security, volume 10355 of Lecture Notes in Computer Science, pages 537–556, Kanazawa, Japan. 2017. Springer, Heidelberg, Germany. DOI: 10.1007/978-3-319-61204-1_27
[CDSV23]
Ignacio Cascudo, Bernardo David, Omer Shlomovits, and Denis Varlakov. Mt. Random: Multi-tiered Randomness Beacons. In Applied Cryptography and Network Security: 21st International Conference, ACNS 2023, Kyoto, Japan, June 19–22, 2023, Proceedings, Part II, pages 645–674, Berlin, Heidelberg. 2023. Springer-Verlag. DOI: 10.1007/978-3-031-33491-7_24
[CGJ+99]
Ran Canetti, Rosario Gennaro, Stanislaw Jarecki, Hugo Krawczyk, and Tal Rabin. Adaptive Security for Threshold Cryptosystems. In Michael J. Wiener, editor, Advances in Cryptology – CRYPTO'99, volume 1666 of Lecture Notes in Computer Science, pages 98–115, Santa Barbara, CA, USA. 1999. Springer, Heidelberg, Germany. DOI: 10.1007/3-540-48405-1_7
[CKPS01]
Christian Cachin, Klaus Kursawe, Frank Petzold, and Victor Shoup. Secure and Efficient Asynchronous Broadcast Protocols. In Joe Kilian, editor, Advances in Cryptology – CRYPTO 2001, volume 2139 of Lecture Notes in Computer Science, pages 524–541, Santa Barbara, CA, USA. 2001. Springer, Heidelberg, Germany. DOI: 10.1007/3-540-44647-8_31
[CKS00]
Christian Cachin, Klaus Kursawe, and Victor Shoup. Random oracles in constantipole: practical asynchronous Byzantine agreement using cryptography (extended abstract). In Gil Neiger, editor, 19th ACM Symposium Annual on Principles of Distributed Computing, pages 123–132, Portland, OR, USA. 2000. Association for Computing Machinery. DOI: 10.1145/343477.343531
[DF90]
Yvo Desmedt and Yair Frankel. Threshold Cryptosystems. In Gilles Brassard, editor, Advances in Cryptology – CRYPTO'89, volume 435 of Lecture Notes in Computer Science, pages 307–315, Santa Barbara, CA, USA. 1990. Springer, Heidelberg, Germany. DOI: 10.1007/0-387-34805-0_28
[DR82]
Danny Dolev and Rüdiger Reischuk. Bounds on Information Exchange for Byzantine Agreement. In Robert L. Probert, Michael J. Fischer, and Nicola Santoro, editors, 1st ACM Symposium Annual on Principles of Distributed Computing, pages 132–140, Ottawa, Canada. 1982. Association for Computing Machinery. DOI: 10.1145/800220.806690
[DS83]
Danny Dolev and H. Raymond Strong. Authenticated algorithms for Byzantine agreement. In SIAM Journal on Computing, volume 12, pages 656–666. 1983. SIAM. DOI: 10.1137/0212045
[DS02]
Paolo D'Arco and Douglas R. Stinson. On Unconditionally Secure Robust Distributed Key Distribution Centers. In Yuliang Zheng, editor, Advances in Cryptology – ASIACRYPT 2002, volume 2501 of Lecture Notes in Computer Science, pages 346–363, Queenstown, New Zealand. 2002. Springer, Heidelberg, Germany. DOI: 10.1007/3-540-36178-2_22
[DXKKR23]
Sourav Das, Zhuolun Xiang, Lefteris Kokoris-Kogias, and Ling Ren. Practical Asynchronous High-threshold Distributed Key Generation and Distributed Polynomial Sampling. In 32nd USENIX Security Symposium (USENIX Security 23), pages 5359–5376, Anaheim, CA. August 2023. USENIX Association.
[DYX+22]
Sourav Das, Thomas Yurek, Zhuolun Xiang, Andrew K. Miller, Lefteris Kokoris-Kogias, and Ling Ren. Practical Asynchronous Distributed Key Generation. In 2022 IEEE Symposium on Security and Privacy, pages 2518–2534, San Francisco, CA, USA. 2022. IEEE Computer Society Press. DOI: 10.1109/SP46214.2022.9833584
[EFR21]
Andreas Erwig, Sebastian Faust, and Siavash Riahi. Large-Scale Non-Interactive Threshold Cryptosystems in the YOSO Model. IACR Cryptol. ePrint Arch., 2021. https://eprint.iacr.org/2021/1290
[Fel87]
Paul Feldman. A Practical Scheme for Non-interactive Verifiable Secret Sharing. In 28th Annual Symposium on Foundations of Computer Science, pages 427–437, Los Angeles, CA, USA. 1987. IEEE Computer Society Press. DOI: 10.1109/SFCS.1987.4
[FG03]
Matthias Fitzi and Juan A Garay. Efficient player-optimal protocols for strong and differential consensus. In Proceedings of the twenty-second annual symposium on Principles of distributed computing (PODC'03), pages 211–220. 2003. DOI: 10.1145/872035.872066
[FH06]
Matthias Fitzi and Martin Hirt. Optimally efficient multi-valued Byzantine agreement. In Eric Ruppert and Dahlia Malkhi, editors, 25th ACM Symposium Annual on Principles of Distributed Computing, pages 163–168, Denver, CO, USA. 2006. Association for Computing Machinery. DOI: 10.1145/1146381.1146407
[FKL18]
Georg Fuchsbauer, Eike Kiltz, and Julian Loss. The algebraic group model and its applications. In Advances in Cryptology (CRYPTO'18): 38th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 19–23, 2018, Proceedings, Part II 38, pages 33–62. 2018. Springer. DOI: 10.1007/978-3-319-96881-0_2
[FLT24]
[FLZL21]
Matthias Fitzi, Chen-Da Liu-Zhang, and Julian Loss. A New Way to Achieve Round-Efficient Byzantine Agreement. In Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing, pages 355–362, New York, NY, USA. 2021. Association for Computing Machinery. DOI: 10.1145/3465084.3467907
[FM88]
Paul Feldman and Silvio Micali. Optimal Algorithms for Byzantine Agreement. In 20th Annual ACM Symposium on Theory of Computing, pages 148–161, Chicago, IL, USA. 1988. ACM Press. DOI: 10.1145/62212.62225
[FM97]
Pesech Feldman and Silvio Micali. An optimal probabilistic protocol for synchronous Byzantine agreement. SIAM Journal on Computing, 26(4):873–933, 1997. DOI: 10.1137/S0097539790187084
[GJKR07]
Rosario Gennaro, Stanislaw Jarecki, Hugo Krawczyk, and Tal Rabin. Secure distributed key generation for discrete-log based cryptosystems. Journal of Cryptology, 2007. DOI: 10.1007/s00145-006-0347-3
[GJM+21]
Kobi Gurkan, Philipp Jovanovic, Mary Maller, Sarah Meiklejohn, Gilad Stern, and Alin Tomescu. Aggregatable distributed key generation. In Annual International Conference on the Theory and Applications of Cryptographic Techniques (EUROCRYPT'21), pages 147–176. 2021. Springer. DOI: 10.1007/978-3-030-77870-5_6
[GKKO07]
Juan A. Garay, Jonathan Katz, Chiu-Yuen Koo, and Rafail Ostrovsky. Round Complexity of Authenticated Broadcast with a Dishonest Majority. In 48th Annual Symposium on Foundations of Computer Science, pages 658–668, Providence, RI, USA. 2007. IEEE Computer Society Press. DOI: 10.1109/FOCS.2007.61
[GLL+22]
Yingzi Gao, Yuan Lu, Zhenliang Lu, Qiang Tang, Jing Xu, and Zhenfeng Zhang. Efficient asynchronous byzantine agreement without private setups. In 2022 IEEE 42nd International Conference on Distributed Computing Systems (ICDCS'22), pages 246–257. 2022. IEEE. DOI: 10.1109/ICDCS54860.2022.00032
[Gro16]
Jens Groth. On the size of pairing-based non-interactive arguments. In Advances in Cryptology–EUROCRYPT 2016: 35th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Vienna, Austria, May 8-12, 2016, Proceedings, Part II 35, pages 305–326. 2016. Springer. DOI: 10.1007/978-3-662-49896-5_11
[Gro21]
[HMQ04]
[HNP05]
Martin Hirt, Jesper Buus Nielsen, and Bartosz Przydatek. Cryptographic Asynchronous Multi-party Computation with Optimal Resilience (Extended Abstract). In Ronald Cramer, editor, Advances in Cryptology – EUROCRYPT 2005, volume 3494 of Lecture Notes in Computer Science, pages 322–340, Aarhus, Denmark. 2005. Springer, Heidelberg, Germany. DOI: 10.1007/11426639_19
[Kat23]
[KG09]
Aniket Kate and Ian Goldberg. Distributed Key Generation for the Internet. In 29th IEEE International Conference on Distributed Computing Systems–ICDCS'09, pages 119-128. 2009. DOI: 10.1109/ICDCS.2009.21
[KGS23]
Chelsea Komlo, Ian Goldberg, and Douglas Stebila. A Formal Treatment of Distributed Key Generation, and New Constructions. IACR Cryptol. ePrint Arch., 2023. https://eprint.iacr.org/2023/292
[KHG12]
Aniket Kate, Yizhou Huang, and Ian Goldberg. Distributed Key Generation in the Wild.. IACR Cryptol. ePrint Arch., 2012:377, 2012. https://eprint.iacr.org/2012/377
[KK06]
Jonathan Katz and Chiu-Yuen Koo. On Expected Constant-Round Protocols for Byzantine Agreement. In Cynthia Dwork, editor, Advances in Cryptology – CRYPTO 2006, volume 4117 of Lecture Notes in Computer Science, pages 445–462, Santa Barbara, CA, USA. 2006. Springer, Heidelberg, Germany. DOI: 10.1007/11818175_27
[KMS20]
Eleftherios Kokoris-Kogias, Dahlia Malkhi, and Alexander Spiegelman. Asynchronous Distributed Key Generation for Computationally-Secure Randomness, Consensus, and Threshold Signatures. In Jay Ligatti, Xinming Ou, Jonathan Katz, and Giovanni Vigna, editors, ACM CCS 2020: 27th Conference on Computer and Communications Security, pages 1751–1767, Virtual Event, USA. 2020. ACM Press. DOI: 10.1145/3372297.3423364
[KZG10]
Aniket Kate, Gregory M. Zaverucha, and Ian Goldberg. Constant-Size Commitments to Polynomials and Their Applications. In Masayuki Abe, editor, Advances in Cryptology – ASIACRYPT 2010, volume 6477 of Lecture Notes in Computer Science, pages 177–194, Singapore. 2010. Springer, Heidelberg, Germany. DOI: 10.1007/978-3-642-17373-8_11
[Lab21]
Torus Lab. Torus: Globally accessible public key infrastructure for everyone. https://tor.us/. 2021.
[LLTW20]
Yuan Lu, Zhenliang Lu, Qiang Tang, and Guiling Wang. Dumbo-MVBA: Optimal Multi-Valued Validated Asynchronous Byzantine Agreement, Revisited. In Yuval Emek and Christian Cachin, editors, 39th ACM Symposium Annual on Principles of Distributed Computing, pages 129–138, Virtual Event, Italy. 2020. Association for Computing Machinery. DOI: 10.1145/3382734.3405707
[Mer88]
Ralph C. Merkle. A Digital Signature Based on a Conventional Encryption Function. In Carl Pomerance, editor, Advances in Cryptology – CRYPTO'87, volume 293 of Lecture Notes in Computer Science, pages 369–378, Santa Barbara, CA, USA. 1988. Springer, Heidelberg, Germany. DOI: 10.1007/3-540-48184-2_32
[Mic16]
Silvio Micali. Byzantine agreement, made trivial. 2016.
[MR21]
Atsuki Momose and Ling Ren. Optimal Communication Complexity of Authenticated Byzantine Agreement. In 35th International Symposium on Distributed Computing (DISC 2021). 2021. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. DOI: 10.4230/LIPIcs.DISC.2021.32
[NBBR16]
Wafa Neji, Kaouther Blibech, and Narjes Ben Rajeb. Distributed key generation protocol with a new complaint management strategy. Security and communication networks, 9(17):4585–4595, 2016. DOI: 10.1002/sec.1651
[Ngu05]
Lan Nguyen. Accumulators from Bilinear Pairings and Applications. In Alfred Menezes, editor, Topics in Cryptology – CT-RSA 2005, volume 3376 of Lecture Notes in Computer Science, pages 275–292, San Francisco, CA, USA. 2005. Springer, Heidelberg, Germany. DOI: 10.1007/978-3-540-30574-3_19
[NRS+20]
Kartik Nayak, Ling Ren, Elaine Shi, Nitin H Vaidya, and Zhuolun Xiang. Improved Extension Protocols for Byzantine Broadcast and Agreement. In Hagit Attiya, editor, 34th International Symposium on Distributed Computing (DISC 2020). 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. DOI: 10.4230/LIPIcs.DISC.2020.28
[Ped91]
Torben P. Pedersen. A Threshold Cryptosystem without a Trusted Party. In Donald W. Davies, editor, Advances in Cryptology – EUROCRYPT'91, volume 547 of Lecture Notes in Computer Science, pages 522–526, Brighton, UK. 1991. Springer, Heidelberg, Germany. DOI: 10.1007/3-540-46416-6_47
[Ped92]
Torben P. Pedersen. Non-Interactive and Information-Theoretic Secure Verifiable Secret Sharing. In Joan Feigenbaum, editor, Advances in Cryptology – CRYPTO'91, volume 576 of Lecture Notes in Computer Science, pages 129–140, Santa Barbara, CA, USA. 1992. Springer, Heidelberg, Germany. DOI: 10.1007/3-540-46766-1_9
[RS60]
Irving S Reed and Gustave Solomon. Polynomial codes over certain finite fields. Journal of the society for industrial and applied mathematics, 8(2):300–304, 1960. DOI: 10.1137/0108018
[SARN20]
Nibesh Shrestha, Ittai Abraham, Ling Ren, and Kartik Nayak. On the Optimality of Optimistic Responsiveness. In Jay Ligatti, Xinming Ou, Jonathan Katz, and Giovanni Vigna, editors, ACM CCS 2020: 27th Conference on Computer and Communications Security, pages 839–857, Virtual Event, USA. 2020. ACM Press. DOI: 10.1145/3372297.3417284
[Sho00]
Victor Shoup. Practical Threshold Signatures. In Bart Preneel, editor, Advances in Cryptology – EUROCRYPT 2000, volume 1807 of Lecture Notes in Computer Science, pages 207–220, Bruges, Belgium. 2000. Springer, Heidelberg, Germany. DOI: 10.1007/3-540-45539-6_15
[SJSW19]
Philipp Schindler, Aljosha Judmayer, Nicholas Stifter, and Edgar Weippl. ETHDKG: Distributed Key Generation with Ethereum Smart Contracts. IACR Cryptol. ePrint Arch., 2019. https://eprint.iacr.org/2019/985
[TCZ+20]
Alin Tomescu, Robert Chen, Yiming Zheng, Ittai Abraham, Benny Pinkas, Guy Golan-Gueta, and Srinivas Devadas. Towards Scalable Threshold Cryptosystems. In 2020 IEEE Symposium on Security and Privacy, pages 877–893, San Francisco, CA, USA. 2020. IEEE Computer Society Press. DOI: 10.1109/SP40000.2020.00059
[TLP22]
Georgios Tsimos, Julian Loss, and Charalampos Papamanthou. Gossiping for Communication-Efficient Broadcast. In Yevgeniy Dodis and Thomas Shrimpton, editors, Advances in Cryptology – CRYPTO 2022, Part III, volume 13509 of Lecture Notes in Computer Science, pages 439–469, Santa Barbara, CA, USA. 2022. Springer, Heidelberg, Germany. DOI: 10.1007/978-3-031-15982-4_15
[YMR+19]
Maofan Yin, Dahlia Malkhi, Michael K. Reiter, Guy Golan-Gueta, and Ittai Abraham. HotStuff: BFT Consensus with Linearity and Responsiveness. In Peter Robinson and Faith Ellen, editors, 38th ACM Symposium Annual on Principles of Distributed Computing, pages 347–356, Toronto, ON, Canada. 2019. Association for Computing Machinery. DOI: 10.1145/3293611.3331591

PDFPDF Open access

History
Submitted: 2024-04-09
Accepted: 2024-06-03
Published: 2024-07-08
How to cite

Nibesh Shrestha, Adithya Bhat, Aniket Kate, and Kartik Nayak, Synchronous Distributed Key Generation without Broadcasts. IACR Communications in Cryptology, vol. 1, no. 2, Jul 08, 2024, doi: 10.62056/ayfhsgvtw.

License

Copyright is held by the author(s)

This work is licensed under a Creative Commons Attribution (CC BY) license.