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Fiat-Shamir for Bounded-Depth Adversaries

Authors Liyan Chen , Yilei Chen , Zikuan Huang , Nuozhou Sun , Tianqi Yang , Yiding Zhang

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ACM Subject Classification
  • Security and privacy → Hash functions and message authentication codes
Keywords
  • Fiat-Shamir
  • Correlation Intractability

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Abstract

We study how to construct hash functions that can securely instantiate the Fiat-Shamir transformation against bounded-depth adversaries. The motivation is twofold. First, given the recent fruitful line of research of constructing cryptographic primitives against bounded-depth adversaries under worst-case complexity assumptions, and the rich applications of Fiat-Shamir, instantiating Fiat-Shamir hash functions against bounded-depth adversaries under worst-case complexity assumptions might lead to further applications (such as SNARG for P, showing the cryptographic hardness of PPAD, etc.) against bounded-depth adversaries. Second, we wonder whether it is possible to overcome the impossibility results of constructing Fiat-Shamir for arguments [Goldwasser, Kalai, FOCS '03] in the setting where the depth of the adversary is bounded, given that the known impossibility results (against p.p.t. adversaries) are contrived.
Our main results give new insights for Fiat-Shamir against bounded-depth adversaries in both the positive and negative directions. On the positive side, for Fiat-Shamir for proofs with certain properties, we show that weak worst-case assumptions are enough for constructing explicit hash functions that give AC⁰[2]-soundness. In particular, we construct an AC⁰[2]-computable correlation-intractable hash family for constant-degree polynomials against AC⁰[2] adversaries, assuming ⊕L/poly ⊈ Sum̃_{n^{-c}}∘AC⁰[2] for some c > 0. This is incomparable to all currently-known constructions, which are typically useful for larger classes and against stronger adversaries, but based on arguably stronger assumptions. Our construction is inspired by the Fiat-Shamir hash function by Peikert and Shiehian [CRYPTO '19] and the fully-homomorphic encryption scheme against bounded-depth adversaries by Wang and Pan [EUROCRYPT '22].
On the negative side, we show Fiat-Shamir for arguments is still impossible to achieve against bounded-depth adversaries. In particular,  
- Assuming the existence of AC⁰[2]-computable CRHF against p.p.t. adversaries, for every poly-size hash function, there is a (p.p.t.-sound) interactive argument that is not AC⁰[2]-sound after applying Fiat-Shamir with this hash function. 
- Assuming the existence of AC⁰[2]-computable CRHF against AC⁰[2] adversaries, there is an AC⁰[2]-sound interactive argument such that for every hash function computable by AC⁰[2] circuits, the argument does not preserve AC⁰[2]-soundness when applying Fiat-Shamir with this hash function. This is a low-depth variant of Goldwasser and Kalai.

Cite As Get BibTex

Liyan Chen, Yilei Chen, Zikuan Huang, Nuozhou Sun, Tianqi Yang, and Yiding Zhang. Fiat-Shamir for Bounded-Depth Adversaries. In 7th Conference on Information-Theoretic Cryptography (ITC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 385, pp. 4:1-4:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITC.2026.4

Author Details

Liyan Chen
  • MIT, Cambridge, MA, USA
Yilei Chen
  • Tsinghua University, Beijing, China
  • Shanghai Qi Zhi Institute, China
Zikuan Huang
  • Shanghai Qi Zhi Institute, China
Nuozhou Sun
  • Carnegie Mellon University, Pittsburgh, PA, USA
Tianqi Yang
  • Columbia University, New York, NY, USA
Yiding Zhang
  • Boston University, MA, USA

Funding

  • Chen, Yilei: Research supported by Tsinghua University startup funding, and Shanghai Qi Zhi Institute Innovation Program.
  • Yang, Tianqi: Research supported in part by NSF Grant CCF-2106429, CCF-2211238, CCF-2212136, CCF-2312224, an Amazon Research Award, and Google CyberNYC.

References

  1. Miklós Ajtai. Generating hard instances of lattice problems (extended abstract). In STOC, pages 99-108, 1996. URL: https://doi.org/10.1145/237814.237838.
  2. Benny Applebaum, Yuval Ishai, and Eyal Kushilevitz. Cryptography in nc^0. SIAM J. Comput., 36(4):845-888, 2006. URL: https://doi.org/10.1137/S0097539705446950.
  3. Boaz Barak. How to go beyond the black-box simulation barrier. In FOCS, pages 106-115. IEEE Computer Society, 2001. URL: https://doi.org/10.1109/SFCS.2001.959885.
  4. Boaz Barak and Oded Goldreich. Universal arguments and their applications. SIAM J. Comput., 38(5):1661-1694, 2008. URL: https://doi.org/10.1137/070709244.
  5. James Bartusek, Liron Bronfman, Justin Holmgren, Fermi Ma, and Ron D. Rothblum. On the (in)security of kilian-based snargs. In Dennis Hofheinz and Alon Rosen, editors, Theory of Cryptography - 17th International Conference, TCC 2019, Nuremberg, Germany, December 1-5, 2019, Proceedings, Part II, volume 11892 of Lecture Notes in Computer Science, pages 522-551. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-36033-7_20.
  6. Nir Bitansky, Yael Tauman Kalai, and Omer Paneth. Multi-collision resistance: a paradigm for keyless hash functions. In STOC, pages 671-684. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188870.
  7. Zvika Brakerski, Venkata Koppula, and Tamer Mour. NIZK from LPN and trapdoor hash via correlation intractability for approximable relations. In Daniele Micciancio and Thomas Ristenpart, editors, Advances in Cryptology - CRYPTO 2020 - 40th Annual International Cryptology Conference, CRYPTO 2020, Santa Barbara, CA, USA, August 17-21, 2020, Proceedings, Part III, volume 12172 of Lecture Notes in Computer Science, pages 738-767. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-56877-1_26.
  8. Matteo Campanelli and Rosario Gennaro. Fine-grained secure computation. In Amos Beimel and Stefan Dziembowski, editors, Theory of Cryptography - 16th International Conference, TCC 2018, Panaji, India, November 11-14, 2018, Proceedings, Part II, volume 11240 of Lecture Notes in Computer Science, pages 66-97. Springer, 2018. URL: https://doi.org/10.1007/978-3-030-03810-6_3.
  9. Ran Canetti, Yilei Chen, Justin Holmgren, Alex Lombardi, Guy N. Rothblum, Ron D. Rothblum, and Daniel Wichs. Fiat-shamir: from practice to theory. In STOC, pages 1082-1090. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316380.
  10. Ran Canetti, Yilei Chen, and Leonid Reyzin. On the correlation intractability of obfuscated pseudorandom functions. In Eyal Kushilevitz and Tal Malkin, editors, Theory of Cryptography - 13th International Conference, TCC 2016-A, Tel Aviv, Israel, January 10-13, 2016, Proceedings, Part I, volume 9562 of Lecture Notes in Computer Science, pages 389-415. Springer, 2016. URL: https://doi.org/10.1007/978-3-662-49096-9_17.
  11. Ran Canetti, Yilei Chen, Leonid Reyzin, and Ron D. Rothblum. Fiat-shamir and correlation intractability from strong kdm-secure encryption. In Jesper Buus Nielsen and Vincent Rijmen, editors, 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 I, volume 10820 of Lecture Notes in Computer Science, pages 91-122. Springer, 2018. URL: https://doi.org/10.1007/978-3-319-78381-9_4.
  12. Ran Canetti, Oded Goldreich, and Shai Halevi. The random oracle methodology, revisited. J. ACM, 51(4):557-594, 2004. URL: https://doi.org/10.1145/1008731.1008734.
  13. Liyan Chen, Yilei Chen, Zikuan Huang, Nuozhou Sun, Tianqi Yang, and Yiding Zhang. Fiat-shamir for bounded-depth adversaries. Cryptology ePrint Archive, Paper 2024/256, 2024. URL: https://eprint.iacr.org/2024/256.
  14. Yilei Chen, Alex Lombardi, Fermi Ma, and Willy Quach. Does fiat-shamir require a cryptographic hash function? In Tal Malkin and Chris Peikert, editors, Advances in Cryptology - CRYPTO 2021 - 41st Annual International Cryptology Conference, CRYPTO 2021, Virtual Event, August 16-20, 2021, Proceedings, Part IV, volume 12828 of Lecture Notes in Computer Science, pages 334-363. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-84259-8_12.
  15. Arka Rai Choudhuri, Sanjam Garg, Abhishek Jain, Zhengzhong Jin, and Jiaheng Zhang. Correlation intractability and snargs from sub-exponential DDH. In Helena Handschuh and Anna Lysyanskaya, editors, Advances in Cryptology - CRYPTO 2023 - 43rd Annual International Cryptology Conference, CRYPTO 2023, Santa Barbara, CA, USA, August 20-24, 2023, Proceedings, Part IV, volume 14084 of Lecture Notes in Computer Science, pages 635-668. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-38551-3_20.
  16. Arka Rai Choudhuri, Pavel Hubácek, Chethan Kamath, Krzysztof Pietrzak, Alon Rosen, and Guy N. Rothblum. Finding a nash equilibrium is no easier than breaking fiat-shamir. In STOC, pages 1103-1114. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316400.
  17. Arka Rai Choudhuri, Abhishek Jain, and Zhengzhong Jin. Non-interactive batch arguments for NP from standard assumptions. In Tal Malkin and Chris Peikert, editors, Advances in Cryptology - CRYPTO 2021 - 41st Annual International Cryptology Conference, CRYPTO 2021, Virtual Event, August 16-20, 2021, Proceedings, Part IV, volume 12828 of Lecture Notes in Computer Science, pages 394-423. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-84259-8_14.
  18. Arka Rai Choudhuri, Abhishek Jain, and Zhengzhong Jin. Snargs for P from LWE. In FOCS, pages 68-79. IEEE, 2021. Google Scholar
  19. Quang Dao, Aayush Jain, and Zhengzhong Jin. Non-interactive zero-knowledge from LPN and MQ. In CRYPTO (9), volume 14928 of Lecture Notes in Computer Science, pages 321-360. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-68400-5_10.
  20. Akshay Degwekar, Vinod Vaikuntanathan, and Prashant Nalini Vasudevan. Fine-grained cryptography. In Matthew Robshaw and Jonathan Katz, editors, Advances in Cryptology - CRYPTO 2016 - 36th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 14-18, 2016, Proceedings, Part III, volume 9816 of Lecture Notes in Computer Science, pages 533-562. Springer, 2016. URL: https://doi.org/10.1007/978-3-662-53015-3_19.
  21. Cynthia Dwork, Moni Naor, Omer Reingold, and Larry J. Stockmeyer. Magic functions. In FOCS, pages 523-534. IEEE Computer Society, 1999. URL: https://doi.org/10.1109/SFFCS.1999.814626.
  22. Shohei Egashira, Yuyu Wang, and Keisuke Tanaka. Fine-grained cryptography revisited. J. Cryptol., 34(3):23, 2021. URL: https://doi.org/10.1007/s00145-021-09390-3.
  23. Amos Fiat and Adi Shamir. How to prove yourself: Practical solutions to identification and signature problems. In Andrew M. Odlyzko, editor, Advances in Cryptology - CRYPTO '86, Santa Barbara, California, USA, 1986, Proceedings, volume 263 of Lecture Notes in Computer Science, pages 186-194. Springer, 1986. URL: https://doi.org/10.1007/3-540-47721-7_12.
  24. Craig Gentry, Amit Sahai, and Brent Waters. Homomorphic encryption from learning with errors: Conceptually-simpler, asymptotically-faster, attribute-based. In Ran Canetti and Juan A. Garay, editors, Advances in Cryptology - CRYPTO 2013 - 33rd Annual Cryptology Conference, Santa Barbara, CA, USA, August 18-22, 2013. Proceedings, Part I, volume 8042 of Lecture Notes in Computer Science, pages 75-92. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40041-4_5.
  25. Craig Gentry and Daniel Wichs. Separating succinct non-interactive arguments from all falsifiable assumptions. In STOC, pages 99-108. ACM, 2011. URL: https://doi.org/10.1145/1993636.1993651.
  26. Shafi Goldwasser and Yael Tauman Kalai. On the (in)security of the fiat-shamir paradigm. In 44th Symposium on Foundations of Computer Science (FOCS 2003), 11-14 October 2003, Cambridge, MA, USA, Proceedings, pages 102-113. IEEE Computer Society, 2003. URL: https://doi.org/10.1109/SFCS.2003.1238185.
  27. Justin Holmgren and Alex Lombardi. Cryptographic hashing from strong one-way functions (or: One-way product functions and their applications). In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 850-858. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00085.
  28. Justin Holmgren, Alex Lombardi, and Ron D. Rothblum. Fiat-shamir via list-recoverable codes (or: parallel repetition of GMW is not zero-knowledge). In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 750-760. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451116.
  29. James Hulett, Ruta Jawale, Dakshita Khurana, and Akshayaram Srinivasan. Snargs for P from sub-exponential DDH and QR. In Orr Dunkelman and Stefan Dziembowski, editors, Advances in Cryptology - EUROCRYPT 2022 - 41st Annual International Conference on the Theory and Applications of Cryptographic Techniques, Trondheim, Norway, May 30 - June 3, 2022, Proceedings, Part II, volume 13276 of Lecture Notes in Computer Science, pages 520-549. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-07085-3_18.
  30. Yuval Ishai and Eyal Kushilevitz. Randomizing polynomials: A new representation with applications to round-efficient secure computation. In FOCS, pages 294-304. IEEE Computer Society, 2000. URL: https://doi.org/10.1109/SFCS.2000.892118.
  31. Abhishek Jain and Zhengzhong Jin. Non-interactive zero knowledge from sub-exponential DDH. In Anne Canteaut and François-Xavier Standaert, editors, Advances in Cryptology - EUROCRYPT 2021 - 40th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Zagreb, Croatia, October 17-21, 2021, Proceedings, Part I, volume 12696 of Lecture Notes in Computer Science, pages 3-32. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-77870-5_1.
  32. Ruta Jawale, Yael Tauman Kalai, Dakshita Khurana, and Rachel Yun Zhang. Snargs for bounded depth computations and PPAD hardness from sub-exponential LWE. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 708-721. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451055.
  33. Yael Tauman Kalai, Alex Lombardi, and Vinod Vaikuntanathan. Snargs and PPAD hardness from the decisional diffie-hellman assumption. In Carmit Hazay and Martijn Stam, editors, Advances in Cryptology - EUROCRYPT 2023 - 42nd Annual International Conference on the Theory and Applications of Cryptographic Techniques, Lyon, France, April 23-27, 2023, Proceedings, Part II, volume 14005 of Lecture Notes in Computer Science, pages 470-498. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-30617-4_16.
  34. Yael Tauman Kalai, Guy N. Rothblum, and Ron D. Rothblum. From obfuscation to the security of fiat-shamir for proofs. In Jonathan Katz and Hovav Shacham, editors, Advances in Cryptology - CRYPTO 2017 - 37th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 20-24, 2017, Proceedings, Part II, volume 10402 of Lecture Notes in Computer Science, pages 224-251. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-63715-0_8.
  35. Yael Tauman Kalai, Vinod Vaikuntanathan, and Rachel Yun Zhang. Somewhere statistical soundness, post-quantum security, and snargs. In Kobbi Nissim and Brent Waters, editors, Theory of Cryptography - 19th International Conference, TCC 2021, Raleigh, NC, USA, November 8-11, 2021, Proceedings, Part I, volume 13042 of Lecture Notes in Computer Science, pages 330-368. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-90459-3_12.
  36. Joe Kilian. A note on efficient zero-knowledge proofs and arguments (extended abstract). In STOC, pages 723-732. ACM, 1992. URL: https://doi.org/10.1145/129712.129782.
  37. Susumu Kiyoshima. Public-coin 3-round zero-knowledge from learning with errors and keyless multi-collision-resistant hash. In CRYPTO (1), volume 13507 of Lecture Notes in Computer Science, pages 444-473. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-15802-5_16.
  38. Alex Lombardi and Vinod Vaikuntanathan. Correlation-intractable hash functions via shift-hiding. In ITCS, volume 215 of LIPIcs, pages 102:1-102:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.ITCS.2022.102.
  39. Silvio Micali. Computationally sound proofs. SIAM J. Comput., 30(4):1253-1298, October 2000. URL: https://doi.org/10.1137/S0097539795284959.
  40. Chris Peikert and Sina Shiehian. Noninteractive zero knowledge for NP from (plain) learning with errors. In Alexandra Boldyreva and Daniele Micciancio, editors, Advances in Cryptology - CRYPTO 2019 - 39th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 18-22, 2019, Proceedings, Part I, volume 11692 of Lecture Notes in Computer Science, pages 89-114. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-26948-7_4.
  41. David Pointcheval and Jacques Stern. Security proofs for signature schemes. In Ueli M. Maurer, editor, Advances in Cryptology - EUROCRYPT '96, International Conference on the Theory and Application of Cryptographic Techniques, Saragossa, Spain, May 12-16, 1996, Proceeding, volume 1070 of Lecture Notes in Computer Science, pages 387-398. Springer, 1996. URL: https://doi.org/10.1007/3-540-68339-9_33.
  42. Yuyu Wang and Jiaxin Pan. Non-interactive zero-knowledge proofs with fine-grained security. In Orr Dunkelman and Stefan Dziembowski, editors, Advances in Cryptology - EUROCRYPT 2022 - 41st Annual International Conference on the Theory and Applications of Cryptographic Techniques, Trondheim, Norway, May 30 - June 3, 2022, Proceedings, Part II, volume 13276 of Lecture Notes in Computer Science, pages 305-335. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-07085-3_11.
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