Abstract
We introduce Adaptive Extractors, which unlike traditional randomness extractors, guarantee security even when an adversary obtains leakage on the source after observing the extractor output. We make a compelling case for the study of such extractors by demonstrating their use in obtaining adaptive leakage in secret sharing schemes.
Specifically, at FOCS 2020, Chattopadhyay, Goodman, Goyal, Kumar, Li, Meka, Zuckerman, built an adaptively secure leakage resilient secret sharing scheme (LRSS) with both rate and leakage rate being \(\mathcal {O}(1/n)\), where \(n\) is the number of parties. In this work, we build an adaptively secure LRSS that offers an interesting trade-off between rate, leakage rate, and the total number of shares from which an adversary can obtain leakage. As a special case, when considering t-out-of-n secret sharing schemes for threshold \(t = \alpha n\) (constant \(0<\alpha <1\)), we build a scheme with a constant rate, constant leakage rate, and allow the adversary leakage from all but \(t-1\) of the shares, while giving her the remaining \(t-1\) shares completely in the clear. (Prior to this, constant rate LRSS scheme tolerating adaptive leakage was unknown for any threshold.)
Finally, we show applications of our techniques to both non-malleable secret sharing and secure message transmission.
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Notes
- 1.
For example, Guruswami and Wooters [22] show that Shamir’s secret sharing scheme is completely insecure when the adversary gets some \(t-1\) shares and just one-bit of leakage from other shares.
- 2.
We note that here we only compare in an adaptive leakage model, without any joint leakage queries on multiple shares (which is called the bounded collusion protocols (BCP) model), for ease of expostion, and discuss the joint model in the technical section later.
- 3.
We note that the original construction of [31] only aimed to achieve non-adaptive security, and we consider a modification, with the aim to expand to adaptive security.
- 4.
Since the leakage queries are adaptive, we require adaptive privacy of the underlying secret sharing scheme, and we show instantiations of the same.
- 5.
\(\mathcal {A}\) is a monotone access structure if for all A, B such that \(A\subset B \subseteq [N]\) and \(A\in \mathcal {A}\), it holds that \(B\in \mathcal {A}\). Throughout this paper whenever we consider a general access structure, we mean a monotone access structure.
- 6.
In [6], the authors refer to adaptive privacy as privacy against dynamic adversaries.
- 7.
q is arbitrary and chosen by \(\mathcal {D}\). It need not be chosen a-priori. We only use it to denote the total number queries made by \(\mathcal {D}\).
- 8.
Note that we only use the re-sampling in proofs and do not require the procedure to be efficient.
- 9.
\(\mathsf {Pre}'\) possibly repeats some information already there in \(\mathsf {Pre}\). For example for \(q=2\), \(s^1\) is there in both \(\mathsf {Pre}\) and \(\mathsf {Pre}'\). It is for the ease of exposition that we have this repetition.
References
Aggarwal, D., Damgård, I., Nielsen, J.B., Obremski, M., Purwanto, E., Ribeiro, J., Simkin, M.: Stronger leakage-resilient and non-malleable secret sharing schemes for general access structures. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11693, pp. 510–539. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26951-7_18
Aggarwal, D., Dodis, Y., Jafargholi, Z., Miles, E., Reyzin, L.: Amplifying privacy in privacy amplification. In: Garay, J.A., Gennaro, R. (eds.) CRYPTO 2014. LNCS, vol. 8617, pp. 183–198. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44381-1_11
Aggarwal, D., Dodis, Y., Kazana, T., Obremski, M.: Non-malleable reductions and applications. In: Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 2015 (2015). https://doi.org/10.1145/2746539.2746544
Aggarwal, D., Dodis, Y., Lovett, S.: Non-malleable codes from additive combinatorics. In: Symposium on Theory of Computing, STOC 2014 (2014). https://doi.org/10.1145/2591796.2591804
Badrinarayanan, S., Srinivasan, A.: Revisiting non-malleable secret sharing. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11476, pp. 593–622. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17653-2_20
Bellare, M., Rogaway, P.: Robust computational secret sharing and a unified account of classical secret-sharing goals. In: Proceedings of the 14th ACM Conference on Computer and Communications Security. CCS 2007, Association for Computing Machinery (2007). https://doi.org/10.1145/1315245.1315268
Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In: Simon, J. (ed.) Proceedings of the 20th Annual ACM Symposium on Theory of Computing, Chicago, Illinois, USA, 2–4 May 1988, pp. 1–10. ACM (1988). https://doi.org/10.1145/62212.62213
Benaloh, J., Leichter, J.: Generalized secret sharing and monotone functions. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 27–35. Springer, New York (1990). https://doi.org/10.1007/0-387-34799-2_3
Benhamouda, F., Degwekar, A., Ishai, Y., Rabin, T.: On the local leakage resilience of linear secret sharing schemes. In: Shacham, H., Boldyreva, A. (eds.) CRYPTO 2018. LNCS, vol. 10991, pp. 531–561. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96884-1_18
Blakley, G.: Safeguarding cryptographic keys. In: Proceedings of the 1979 AFIPS National Computer Conference. AFIPS Press (1979)
Brian, G., Faonio, A., Obremski, M., Simkin, M., Venturi, D.: Non-malleable secret sharing against bounded joint-tampering attacks in the plain model. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12172, pp. 127–155. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56877-1_5
Brian, G., Faonio, A., Venturi, D.: Continuously non-malleable secret sharing for general access structures. In: Hofheinz, D., Rosen, A. (eds.) TCC 2019. LNCS, vol. 11892, pp. 211–232. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-36033-7_8
Chattopadhyay, E., et al.: Extractors and secret sharing against bounded collusion protocols. In: 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 (2020). https://doi.org/10.1109/FOCS46700.2020.00117
Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols (extended abstract). In: Simon, J. (ed.) Proceedings of the 20th Annual ACM Symposium on Theory of Computing, Chicago, Illinois, USA, 2–4 May 1988, pp. 11–19. ACM (1988). https://doi.org/10.1145/62212.62214
Davì, F., Dziembowski, S., Venturi, D.: Leakage-resilient storage. In: 7th International Conference on Security and Cryptography for Networks, SCN 2010 (2010). https://doi.org/10.1007/978-3-642-15317-4_9
Dodis, Y., Ostrovsky, R., Reyzin, L., Smith, A.: Fuzzy extractors: how to generate strong keys from biometrics and other noisy data. SIAM J. Comput. 38(1), 97–139 (2008), arXiv:cs/0602007
Dziembowski, S., Pietrzak, K.: Intrusion-resilient secret sharing. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007 (2007). https://doi.org/10.1109/FOCS.2007.35
Faonio, A., Venturi, D.: Non-malleable secret sharing in the computational setting: adaptive tampering, noisy-leakage resilience, and improved rate. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11693, pp. 448–479. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26951-7_16
Faust, S., Rabin, T., Reyzin, L., Tromer, E., Vaikuntanathan, V.: Protecting circuits from leakage: the computationally-bounded and noisy cases. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 135–156. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-13190-5_7
Goyal, V., Kumar, A.: Non-malleable secret sharing. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018 (2018). https://doi.org/10.1145/3188745.3188872
Guruswami, V., Umans, C., Vadhan, S.P.: Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. In: IEEE Conference on Computational Complexity, pp. 96–108 (2007)
Guruswami, V., Wootters, M.: Repairing reed-solomon codes. In: Proceedings of the Forty-eighth Annual ACM Symposium on Theory of Computing. STOC 2016. ACM, New York (2016). https://doi.org/10.1145/2897518.2897525
Ishai, Y., Sahai, A., Wagner, D.: Private circuits: securing hardware against probing attacks. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 463–481. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45146-4_27
Kocher, P.C.: Timing attacks on implementations of Diffie-Hellman, RSA, DSS, and other systems. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 104–113. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-68697-5_9
Kumar, A., Meka, R., Sahai, A.: Leakage-resilient secret sharing against colluding parties. In: 60th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2019 (2019). https://doi.org/10.1109/FOCS.2019.00045
Lin, F., Cheraghchi, M., Guruswami, V., Safavi-Naini, R., Wang, H.: Non-malleable secret sharing against affine tampering. CoRR abs/1902.06195 (2019). http://arxiv.org/abs/1902.06195
Liu, F.-H., Lysyanskaya, A.: Tamper and leakage resilience in the split-state model. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 517–532. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_30
Nisan, N., Zuckerman, D.: Randomness is linear in space. J. Comput. Syst. Sci. 52(1), 43–53 (1996)
Rothblum, G.N.: How to compute under \({\cal{AC}}^{\sf 0}\) leakage without secure hardware. In: Safavi-Naini, R., Canetti, R. (eds.) CRYPTO 2012. LNCS, vol. 7417, pp. 552–569. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32009-5_32
Shamir, A.: How to share a secret. Commun. ACM 22(11), 612–613 (1979)
Srinivasan, A., Vasudevan, P.N.: Leakage resilient secret sharing and applications. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11693, pp. 480–509. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26951-7_17
Vadhan, S.: Pseudorandomness. Foundations and Trends in Theoretical Computer Science. Now Publishers (2012). http://people.seas.harvard.edu/~salil/pseudorandomness/
Zimand, M.: Exposure-resilient extractors. In: 21st Annual IEEE Conference on Computational Complexity (CCC 2006). IEEE Computer Society (2006). https://doi.org/10.1109/CCC.2006.19
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We thank all the anonymous reviewers who provided their valuable comments on an earlier version of this manuscript.
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Chandran, N., Kanukurthi, B., Obbattu, S.L.B., Sekar, S. (2021). Adaptive Extractors and Their Application to Leakage Resilient Secret Sharing. In: Malkin, T., Peikert, C. (eds) Advances in Cryptology – CRYPTO 2021. CRYPTO 2021. Lecture Notes in Computer Science(), vol 12827. Springer, Cham. https://doi.org/10.1007/978-3-030-84252-9_20
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