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ZKBdf: A ZKBoo-Based Quantum-Secure Verifiable Delay Function with Prover-Secret

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Applied Cryptography and Network Security Workshops (ACNS 2023)

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Abstract

Since the formalization of Verifiable Delay Functions (VDF) by Boneh et al. in 2018, VDFs have been adopted for use in blockchain consensus protocols and random beacon implementations. However, the impending threat to VDF-based applications comes in the form of Shor’s algorithm running on quantum computers in the future which can break the discrete logarithm and integer factorization problems that existing VDFs are based on. Clearly, there is a need for quantum-secure VDFs. In this paper, we propose ZKBdf, which makes use of ZKBoo, a zero-knowledge proof system for verifiable computation, as the basis for realizing a quantum-secure VDF. We describe the algorithm, provide the security proofs, implement the scheme and measure the execution and size requirements. In addition, as ZKBdf extends the standard VDF with an extra “Prover-secret” feature, new VDF use-cases are also explored.

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Notes

  1. 1.

    Source codes at https://github.com/tanteikg/zkbdf.

  2. 2.

    We define \(ZKBdf.VerifyPseudo(H(e_k),Res,\)z\(,Cha,T) \rightarrow (result \in \{1,0\})\) as a function run by the Verifier. The difference with zkbdf.Verify is that the input proof is the entire set of zkboo proofs z instead of the PCP proofs \(\pi \). As a reference, zkbdf.VerifyPseudo achieves the same completeness, soundness, sequentiality, uniqueness, and quantum-secure properties. It only does not achieve the efficient-verifiability property to make it a VDF.

  3. 3.

    The quantum-secure VDF by Chavez-Saab et al. [13] lacks published implementation details for comparison.

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Acknowledgement

Jianying Zhou is supported by A*STAR under its RIE2020 Advanced Manufacturing and Engineering (AME) Industry Alignment Fund - Pre Positioning (IAF-PP) Award A19D6a0053. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of A*STAR. Zengpeng Li is supported by the Natural Science Foundation of Shandong Province, China. (grant No. ZR2023MF045) and the Natural Science Foundation of Qingdao, China (grant No. 23-2-1-152-zyyd-jch)

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Authors and Affiliations

  1. pQCee Pte Ltd, Singapore, Singapore

    Teik Guan Tan

  2. Singapore University of Technology and Design, Singapore, Singapore

    Pawel Szalachowski & Jianying Zhou

  3. Queen’s University Belfast, Belfast, UK

    Vishal Sharma

  4. School of Cyber Science and Technology, Shandong University, Jinan, China

    Zeng Peng Li

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  1. Teik Guan Tan
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  2. Vishal Sharma
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  3. Zeng Peng Li
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  4. Pawel Szalachowski
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  5. Jianying Zhou
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Correspondence to Teik Guan Tan .

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Editors and Affiliations

  1. Singapore University of Technology and Design, Singapore, Singapore

    Jianying Zhou

  2. Radboud University Nijmegen, Nijmegen, The Netherlands

    Lejla Batina

  3. Shandong University of Science and Technology, Qingdao, China

    Zengpeng Li

  4. University of Science and Technology of China, Hefei, China

    Jingqiang Lin

  5. University of Padua, Padua, Italy

    Eleonora Losiouk

  6. Concordia University, Montreal, QC, Canada

    Suryadipta Majumdar

  7. Illinois at Singapore Pte Ltd., Singapore, Singapore

    Daisuke Mashima

  8. Technical University Denmark, Kongens Lyngby, Denmark

    Weizhi Meng

  9. Radboud University Nijmegen, Nijmegen, The Netherlands

    Stjepan Picek

  10. Florida International University, Miami, FL, USA

    Mohammad Ashiqur Rahman

  11. Zhejiang Gongshang University, Hangzhou, China

    Jun Shao

  12. SECOM Co., Ltd., University of Tsukuba, Tokyo / Ibaraki, Japan

    Masaki Shimaoka

  13. Royal Holloway University of London, Egham, UK

    Ezekiel Soremekun

  14. University of Aizu, Aizuwakamatsu, Fukushima, Japan

    Chunhua Su

  15. Universiti Sains Malaysia, Gelugor, Malaysia

    Je Sen Teh

  16. University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Aleksei Udovenko

  17. City University of Hong Kong, Hong Kong, Hong Kong

    Cong Wang

  18. Griffith University, Southport, QLD, Australia

    Leo Zhang

  19. Delft University of Technology, Delft, The Netherlands

    Yury Zhauniarovich

Appendix - ZKBdf Application Areas

Appendix - ZKBdf Application Areas

At present, we see the use of VDFs in consensus protocols for blockchains such as Ethereum (ethereum.org), Tezos (tezos.foundation) and Chia (chia.net) as well as in constructing time-stamping services and random beacons [21, 34, 50]. However, we believe that there is a wider use-case for VDFs if the functionality of a Prover-secret is included. In this appendix, we take an exploratory approach to identify other possible use-cases where applications can use ZKBdf to improve outcomes. These are described below.

1.1 A.1 Limiting Authentication Retries

We find a use-case where a delay function is needed during the authentication process to limit a brute-force attack against a backend authentication service. Broken Authentication is amongst the top 10 security risks highlighted by OWASP (Open Web Application Security Project) [42] where one of the ways to address such risks is to introduce an increasing delay for repeated failed authentication attempts. Such a setup, however, requires the backend authentication service to maintain failed authentication states for every user which inadvertently adds resource overheads and complexity especially in distributed systems. There are also many protocols such as Bitcoin [41], Transport Layer Security [45] and Wi-FI Protected Access (IEEE 802.11-2020) which do not require tracking of failed authentication attempts. A stateless delay mechanism using client-side puzzles is presented by Aura et al [3] where every authentication is preceded with a PoW challenge which the authentication client needs to solve, before the server verifies the solution. Similar mechanisms are also used to prevent brute-force denial-of-service network attacks and limiting peer-to-peer sybil attacks [18].

The advantage of using ZKBdf instead of a client-side PoW puzzle is that the number of authentication retries that a hacker can make is deterministic and no longer dependent on the amount of resources available to the hacker. Increasing the amount of CPU/memory resources at the hacker’s end does not increase the number of authentication retries, and this will serve to deter hackers while not increasing the carbon footprint caused by ever-more complex puzzles.

1.2 A.2 Improving Auction Liveliness

In a classical English auction, an item is put on offer for participants to bid in an open outcry manner. The auctioneer asks for participants to place bids higher than the previous bid, and when an elapsed period has occurred without any participants placing any higher bids, the auction is closed with the winner being the participant who submitted the latest (and highest) bid. Online auctions that happen on the Internet, on the other hand, mostly do not have a concept of an elapsed time since the last bid. Instead, there is auction end-time whereby the highest bid received before the end-time is the winning bid. This has given rise to situations where participants are passive throughout most of the auction period and are only active at the closing moments of the auction where bid sniping [47] occurs.

The ZKBdf protocol could be used to allow honest participants to determine the end of the auction prior to the end-time. When a participant submits a valid highest bid, the participant is issued with a VDF challenge which effectively starts the elapsed time computation. If no higher bid is received prior to the participant completing the VDF challenge and submitting the response, then the participant would have won the auction, thus ending the auction. The advantage of using the VDF here is that we expect participants will no longer wait till the closing moments before bidding. Another possible positive outcome would be the detection of shill bidding (an agent working for a corrupt seller to bid up the price without any intention to buy) since the seller’s agent is unlikely to submit the VDF response to close the auction.

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Tan, T.G., Sharma, V., Li, Z.P., Szalachowski, P., Zhou, J. (2023). ZKBdf: A ZKBoo-Based Quantum-Secure Verifiable Delay Function with Prover-Secret. In: Zhou, J., et al. Applied Cryptography and Network Security Workshops. ACNS 2023. Lecture Notes in Computer Science, vol 13907. Springer, Cham. https://doi.org/10.1007/978-3-031-41181-6_29

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