Abstract
Property-preserving hash functions allow for compressing long inputs \(x_0\) and \(x_1\) into short hashes \(h(x_0)\) and \(h(x_1)\) in a manner that allows for computing a predicate \(P(x_0, x_1)\) given only the two hash values without having access to the original data. Such hash functions are said to be adversarially robust if an adversary that gets to pick \(x_0\) and \(x_1\) after the hash function has been sampled, cannot find inputs for which the predicate evaluated on the hash values outputs the incorrect result.
In this work we construct robust property-preserving hash functions for the hamming-distance predicate which distinguishes inputs with a hamming distance at least some threshold t from those with distance less than t. The security of the construction is based on standard lattice hardness assumptions.
Our construction has several advantages over the best known previous construction by Fleischhacker and Simkin (Eurocrypt 2021). Our construction relies on a single well-studied hardness assumption from lattice cryptography whereas the previous work relied on a newly introduced family of computational hardness assumptions. In terms of computational effort, our construction only requires a small number of modular additions per input bit, whereas the work of Fleischhacker and Simkin required several exponentiations per bit as well as the interpolation and evaluation of high-degree polynomials over large fields. An additional benefit of our construction is that the description of the hash function can be compressed to \(\lambda \) bits assuming a random oracle. Previous work has descriptions of length \(\mathcal {O}({\ell \lambda })\) bits for input bit-length \(\ell \).
We prove a lower bound on the output size of any property-preserving hash function for the hamming distance predicate. The bound shows that the size of our hash value is not far from optimal.
N. Fleischhacker—Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2092 CASA - 390781972.
K. Green Larsen—Supported by Independent Research Fund Denmark (DFF) Sapere Aude Research Leader Grant No. 9064-00068B.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
We do not care about the exact size of their gap, since we will focus on a strictly stronger predicate in this work.
References
Aggarwal, D., Dadush, D., Regev, O., Stephens-Davidowitz, N.: Solving the shortest vector problem in \(2^n\) time using discrete Gaussian sampling: extended abstract. In: Servedio, R.A., Rubinfeld, R. (eds.) 47th Annual ACM Symposium on Theory of Computing, pp. 733–742. ACM Press, Portland, OR, USA (2015). https://doi.org/10.1145/2746539.2746606
Aggarwal, D., Li, J., Nguyen, P.Q., Stephens-Davidowitz, N.: Slide reduction, revisited—Filling the gaps in SVP approximation. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12171, pp. 274–295. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56880-1_10
Aggarwal, D., Stephens-Davidowitz, N.: Just take the average! An embarrassingly simple \(2^n\)-time algorithm for SVP (and CVP). In: Seidel, R. (ed.) 1st Symposium on Simplicity in Algorithms (SOSA 2018). OpenAccess Series in Informatics (OASIcs), vol. 61, pp. 12:1–12:19. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2018). https://doi.org/10.4230/OASIcs.SOSA.2018.12
Ajtai, M.: Generating hard instances of lattice problems (extended abstract). In: 28th Annual ACM Symposium on Theory of Computing, pp. 99–108. ACM Press, Philadelphia, PA, USA (1996). https://doi.org/10.1145/237814.237838
Alon, N., Matias, Y., Szegedy, M.: The space complexity of approximating the frequency moments. In: 28th Annual ACM Symposium on Theory of Computing, pp. 20–29. ACM Press, Philadelphia, PA, USA (1996). https://doi.org/10.1145/237814.237823
Ben-Eliezer, O., Jayaram, R., Woodruff, D.P., Yogev, E.: A framework for adversarially robust streaming algorithms. In: Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pp. 63–80 (2020)
Ben-Eliezer, O., Yogev, E.: The adversarial robustness of sampling. In: Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, pp. 49–62 (2020)
Bloom, B.H.: Space/time trade-offs in hash coding with allowable errors. Commun. ACM 13(7), 422–426 (1970)
Boyle, E., LaVigne, R., Vaikuntanathan, V.: Adversarially robust property-preserving hash functions. In: Blum, A. (ed.) ITCS 2019: 10th Innovations in Theoretical Computer Science Conference, vol. 124, pp. 16:1–16:20. LIPIcs, San Diego, CA, USA (2019). https://doi.org/10.4230/LIPIcs.ITCS.2019.16
Clayton, D., Patton, C., Shrimpton, T.: Probabilistic data structures in adversarial environments. In: Cavallaro, L., Kinder, J., Wang, X., Katz, J. (eds.) ACM CCS 2019: 26th Conference on Computer and Communications Security, pp. 1317–1334. ACM Press (2019). https://doi.org/10.1145/3319535.3354235
Donoho, D.L.: Compressed sensing. IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006)
Fleischhacker, N., Simkin, M.: Robust property-preserving hash functions for hamming distance and more. In: Canteaut, A., Standaert, F.-X. (eds.) EUROCRYPT 2021. LNCS, vol. 12698, pp. 311–337. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-77883-5_11
Goodrich, M.T., Mitzenmacher, M.: Invertible bloom lookup tables. In: 2011 49th Annual Allerton Conference on Communication, Control, and Computing (Allerton), pp. 792–799. IEEE (2011)
Hardt, M., Woodruff, D.P.: How robust are linear sketches to adaptive inputs? In: Boneh, D., Roughgarden, T., Feigenbaum, J. (eds.) 45th Annual ACM Symposium on Theory of Computing, pp. 121–130. ACM Press, Palo Alto, CA, USA (2013). https://doi.org/10.1145/2488608.2488624
Indyk, P., Motwani, R.: Approximate nearest neighbors: towards removing the curse of dimensionality. In: 30th Annual ACM Symposium on Theory of Computing, pp. 604–613. ACM Press, Dallas, TX, USA (1998). https://doi.org/10.1145/276698.276876
Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)
Micciancio, D., Peikert, C.: Hardness of SIS and LWE with small parameters. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8042, pp. 21–39. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_2
Miltersen, P.B., Nisan, N., Safra, S., Wigderson, A.: On data structures and asymmetric communication complexity. J. Comput. Syst. Sci. 57(1), 37–49 (1998)
Mironov, I., Naor, M., Segev, G.: Sketching in adversarial environments. In: Ladner, R.E., Dwork, C. (eds.) 40th Annual ACM Symposium on Theory of Computing, pp. 651–660. ACM Press, Victoria, BC, Canada (2008). https://doi.org/10.1145/1374376.1374471
Muthukrishnan, S.: Data streams: algorithms and applications. In: 14th Annual ACM-SIAM Symposium on Discrete Algorithms, p. 413. ACM-SIAM, Baltimore, MD, USA (2003)
Naor, M., Yogev, E.: Bloom filters in adversarial environments. In: Gennaro, R., Robshaw, M. (eds.) CRYPTO 2015. LNCS, vol. 9216, pp. 565–584. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-48000-7_28
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 International Association for Cryptologic Research
About this paper
Cite this paper
Fleischhacker, N., Larsen, K.G., Simkin, M. (2022). Property-Preserving Hash Functions for Hamming Distance from Standard Assumptions. In: Dunkelman, O., Dziembowski, S. (eds) Advances in Cryptology – EUROCRYPT 2022. EUROCRYPT 2022. Lecture Notes in Computer Science, vol 13276. Springer, Cham. https://doi.org/10.1007/978-3-031-07085-3_26
Download citation
DOI: https://doi.org/10.1007/978-3-031-07085-3_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-07084-6
Online ISBN: 978-3-031-07085-3
eBook Packages: Computer ScienceComputer Science (R0)
