Abstract
Non-malleable Codes give us the following property: their codewords cannot be tampered into codewords of related messages. Privacy Amplification allows parties to convert their weak shared secret into a fully hidden, uniformly distributed secret key, while communicating on a fully tamperable public channel. In this work, we show how to construct a constant round privacy amplification protocol from any augmented split-state non-malleable code. Existentially, this gives us another primitive (in addition to optimal non-malleable extractors) whose optimal construction would solve the long-standing open problem of building constant round privacy amplification with optimal entropy loss and min-entropy requirement. Instantiating our code with the current best known NMC gives us an 8-round privacy amplification protocol with entropy loss \(\mathcal {O}(\log (n)+ \kappa \log (\kappa ))\) and min-entropy requirement \(\varOmega (\log (n) +\kappa \log (\kappa ))\), where \(\kappa \) is the security parameter and n is the length of the shared weak secret. In fact, for our result, even the weaker primitive of Non-malleable Randomness Encoders suffice.
We view our result as an exciting connection between two of the most fascinating and well-studied information theoretic primitives, non-malleable codes and privacy amplification.
E. Chattopadhyay—Research supported by NSF grant CCF-1412958 and the Simons foundation.
B. Kanukurthi—Research supported in part by Department of Science and Technology Inspire Faculty Award.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
We can instantiate our construction with recent NMC constructions like [4, 31]. We wish to point out that while using [31] slightly improves the entropy loss here (not optimal though), using the constant rate NMC of [4] or [31] results in more entropy loss. This is because the error of their NMC is worse than [30].
- 2.
Note that the non-malleable secret sharing schemes using split-state NMCs achieve non-malleability in the independent tampering model, not an arbitrary tampering.
- 3.
Here \((f,g)(\mathsf {NMREnc}_2(\mathcal {R}))\) just denotes the tampering by the split-state tampering functions f and g on the corresponding states.
- 4.
For simplicity in the proof, we may assume here that the decoder \(\mathsf {Dec}\) never outputs \(\bot \). This can be done by replacing \(\bot \) with some fixed string, like 00..0.
References
Aggarwal, D., Agrawal, S., Gupta, D., Maji, H.K., Pandey, O., Prabhakaran, M.: Optimal computational split-state non-malleable codes. In: Kushilevitz, E., Malkin, T. (eds.) TCC 2016, Part II. LNCS, vol. 9563, pp. 393–417. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49099-0_15
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, Portland, OR, USA, 14–17 June 2015, pp. 459–468 (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, New York, NY, USA, 31 May–03 June 2014, pp. 774–783 (2014). https://doi.org/10.1145/2591796.2591804
Aggarwal, D., Obremski, M.: Inception makes non-malleable codes shorter as well! IACR Cryptology ePrint Archive 2019/399 (2019). https://eprint.iacr.org/2019/399
Agrawal, S., Gupta, D., Maji, H.K., Pandey, O., Prabhakaran, M.: A rate-optimizing compiler for non-malleable codes against bit-wise tampering and permutations. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015, Part I. LNCS, vol. 9014, pp. 375–397. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46494-6_16
Bennett, C., Brassard, G., Robert, J.M.: Privacy amplification by public discussion. SIAM J. Comput. 17(2), 210–229 (1988)
Bennett, C.H., Brassard, G., Crépeau, C., Maurer, U.M.: Generalized privacy amplification. IEEE Trans. Inf. Theory 41(6), 1915–1923 (1995)
Chandran, N., Kanukurthi, B., Ostrovsky, R., Reyzin, L.: Privacy amplification with asymptotically optimal entropy loss. In: Schulman, L.J. (ed.) Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5–8 June 2010, pp. 785–794. ACM (2010). https://doi.org/10.1145/1806689.1806796
Chattopadhyay, E., Goyal, V., Li, X.: Non-malleable extractors and codes, with their many tampered extensions. In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, 18–21 June 2016, pp. 285–298 (2016). https://doi.org/10.1145/2897518.2897547
Chattopadhyay, E., Kanukurthi, B., Obbattu, S.L.B., Sekar, S.: Privacy amplification from non-malleable codes. Cryptology ePrint Archive, Report 2018/293 (2018). https://eprint.iacr.org/2018/293
Chattopadhyay, E., Li, X.: Explicit non-malleable extractors, multi-source extractors, and almost optimal privacy amplification protocols. In: IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9–11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pp. 158–167 (2016). https://doi.org/10.1109/FOCS.2016.25
Chattopadhyay, E., Zuckerman, D.: Non-malleable codes against constant split-state tampering. In: 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, 18–21 October 2014, pp. 306–315 (2014). https://doi.org/10.1109/FOCS.2014.40
Cheraghchi, M., Guruswami, V.: Non-malleable coding against bit-wise and split-state tampering. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 440–464. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54242-8_19
Cohen, G.: Making the most of advice: new correlation breakers and their applications. In: IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9–11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pp. 188–196 (2016). https://doi.org/10.1109/FOCS.2016.28
Cohen, G., Raz, R., Segev, G.: Non-malleable extractors with short seeds and applications to privacy amplification. In: Proceedings of the 27th Conference on Computational Complexity, CCC 2012, Porto, Portugal, 26–29 June 2012, pp. 298–308 (2012). https://doi.org/10.1109/CCC.2012.21
Dodis, Y., Spencer, J.: On the (non-)universality of the one-time pad. In: 43rd Annual Symposium on Foundations of Computer Science, pp. 376–385. IEEE (2002)
Dodis, Y., Katz, J., Reyzin, L., Smith, A.: Robust fuzzy extractors and authenticated key agreement from close secrets. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 232–250. Springer, Heidelberg (2006). https://doi.org/10.1007/11818175_14
Dodis, Y., Li, X., Wooley, T.D., Zuckerman, D.: Privacy amplification and non-malleable extractors via character sums. In: Ostrovsky, R. (ed.) IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Palm Springs, CA, USA, 22–25 October 2011, pp. 668–677. IEEE (2011). https://doi.org/10.1109/FOCS.2011.67
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
Dodis, Y., Wichs, D.: Non-malleable extractors and symmetric key cryptography from weak secrets. In: Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, pp. 601–610, Bethesda, Maryland, 31 May–2 Jun 2009 (2009)
Dziembowski, S., Pietrzak, K., Wichs, D.: Non-malleable codes. In: Innovations in Computer Science - ICS 2010, Proceedings, Tsinghua University, Beijing, China, 5–7 January 2010, pp. 434–452 (2010). http://conference.itcs.tsinghua.edu.cn/ICS2010/content/papers/34.html
Goyal, V., Kumar, A.: Non-malleable secret sharing. IACR Cryptology ePrint Archive 2018/316 (2018). https://eprint.iacr.org/2018/316
Goyal, V., Pandey, O., Richelson, S.: Textbook non-malleable commitments. In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, 18–21 June 2016, pp. 1128–1141 (2016). https://doi.org/10.1145/2897518.2897657
Kanukurthi, B., Obbattu, S.L.B., Sekar, S.: Four-state non-malleable codes with explicit constant rate. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017, Part II. LNCS, vol. 10678, pp. 344–375. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70503-3_11
Kanukurthi, B., Obbattu, S.L.B., Sekar, S.: Non-malleable randomness encoders and their applications. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018, Part III. LNCS, vol. 10822, pp. 589–617. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78372-7_19
Kanukurthi, B., Reyzin, L.: Key agreement from close secrets over unsecured channels. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 206–223. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-01001-9_12
Li, X.: Design extractors, non-malleable condensers and privacy amplification. In: Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19–22, 2012. pp. 837–854 (2012). https://doi.org/10.1145/2213977.2214052, https://doi.org/10.1145/2213977.2214052
Li, X.: Non-malleable extractors, two-source extractors and privacy amplification. In: 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, 20–23 October 2012, pp. 688–697 (2012). https://doi.org/10.1109/FOCS.2012.26
Li, X.: Non-malleable Condensers for arbitrary min-entropy, and almost optimal protocols for privacy amplification. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015, Part I. LNCS, vol. 9014, pp. 502–531. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46494-6_21
Li, X.: Improved non-malleable extractors, non-malleable codes and independent source extractors. In: Symposium on Theory of Computing, STOC 2017, Montreal, Canada, 19–23 June 2017 (2017)
Li, X.: Non-malleable extractors and non-malleable codes: partially optimal constructions. In: Computational Complexity Conference, CCC 2019, New Brunswick, 18–20 June 2019 (2019)
Maurer, U.M.: Protocols for secret key agreement by public discussion based on common information. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 461–470. Springer, Heidelberg (1993). https://doi.org/10.1007/3-540-48071-4_32
Maurer, U., Wolf, S.: Privacy amplification secure against active adversaries. In: Kaliski, B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 307–321. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0052244
Nisan, N., Zuckerman, D.: Randomness is linear in space. J. Comput. Syst. Sci. 52(1), 43–53 (1996)
Renner, R., Wolf, S.: Unconditional authenticity and privacy from an arbitrarily weak secret. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 78–95. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45146-4_5
Srinivasan, A.: Personal communication
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Chattopadhyay, E., Kanukurthi, B., Obbattu, S.L.B., Sekar, S. (2019). Privacy Amplification from Non-malleable Codes. In: Hao, F., Ruj, S., Sen Gupta, S. (eds) Progress in Cryptology – INDOCRYPT 2019. INDOCRYPT 2019. Lecture Notes in Computer Science(), vol 11898. Springer, Cham. https://doi.org/10.1007/978-3-030-35423-7_16
Download citation
DOI: https://doi.org/10.1007/978-3-030-35423-7_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-35422-0
Online ISBN: 978-3-030-35423-7
eBook Packages: Computer ScienceComputer Science (R0)