Abstract
There are natural cryptographic applications where an adversary only gets to tamper a high-speed data stream on the fly based on her view so far, namely, the lookahead tampering model. Since the adversary can easily substitute transmitted messages with her messages, it is farfetched to insist on strong guarantees like error-correction or, even, manipulation detection. Dziembowski, Pietrzak, and Wichs (ICS–2010) introduced the notion of non-malleable codes that provide a useful message integrity for such scenarios. Intuitively, a non-malleable code ensures that the tampered codeword encodes the original message or a message that is entirely independent of the original message.
Our work studies the following tampering model. We encode a message into \(k\geqslant 1\) secret shares, and we transmit each share as a separate stream of data. Adversaries can perform lookahead tampering on each share, albeit, independently. We call this k-lookahead model.
First, we show a hardness result for the k-lookahead model. To transmit an \(\ell \)-bit message, the cumulative length of the secret shares must be at least \(\frac{k}{k-1}\ell \). This result immediately rules out the possibility of a solution with \(k=1\). Next, we construct a solution for 2-lookahead model such that the total length of the shares is \(3\ell \), which is only 1.5x of the optimal encoding as indicated by our hardness result.
Prior work considers stronger model of split-state encoding that creates \(k\geqslant 2\) secret shares, but protects against adversaries who perform arbitrary (but independent) tampering on each secret share. The size of the secret shares of the most efficient 2-split-state encoding is \(\ell \log \ell /\log \log \ell \) (Li, ECCC–2018). Even though k-lookahead is a weaker tampering class, our hardness result matches that of k-split-state tampering by Cheraghchi and Guruswami (TCC–2014). However, our explicit constructions above achieve much higher efficiency in encoding.
H. K. Maji—The research effort is supported in part by an NSF CRII Award CNS–1566499, an NSF SMALL Award CNS–1618822, and an REU CNS–1724673.
H. K. Maji and M. Wang—The research effort is supported in part by a Purdue Research Foundation grant.
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.
Concurrent and independent work of [21] obtained similar result.
- 2.
- 3.
Specifically, in their proof of Theorem 5.3, they picked two messages \(s_0\), \(s_1\) along with \(X_\eta \) that satisfy the property we require for \(m_0\), \(m_1\) in the imported lemma. Also, we stress that their proof not only showed \(s_0\) and \(s_1\) exist, but there are multiple choices for the pair. This gives us the freedom when we pick our \(m_0\) and \(m_1\). We make use of this in our proof.
- 4.
We note that such codewords would exist otherwise we can show that the last bit of the first state is redundant for decoding. This way we can obtain a smaller encoding. Then, w.l.o.g., we can apply our argument on this new encoding.
- 5.
Specifically, Theorem 30 in [2] states that there exists a constant-rate non-malleable reduction from 2-split-state tampering family to the following tampering function family consisting of union of split-state lookahead and forgetful tampering functions.
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. 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: Servedio, R.A., Rubinfeld, R. (eds.) 47th Annual ACM Symposium on Theory of Computing, Portland, OR, USA, 14–17 June 2015, pp. 459–468. ACM Press (2015)
Aggarwal, D., Dodis, Y., Lovett, S.: Non-malleable codes from additive combinatorics. In: Shmoys, D.B. (ed.) 46th Annual ACM Symposium on Theory of Computing, New York, NY, USA, 31 May–3 June 2014, pp. 774–783. ACM Press (2014)
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. LNCS, vol. 9014, pp. 375–397. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46494-6_16
Ball, M., Dachman-Soled, D., Kulkarni, M., Malkin, T.: Non-malleable codes for bounded depth, bounded fan-in circuits. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 881–908. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-49896-5_31
Chandran, N., Goyal, V., Mukherjee, P., Pandey, O., Upadhyay, J.: Block-wise non-malleable codes. In: Chatzigiannakis, I., Mitzenmacher, M., Rabani, Y., Sangiorgi, D. (eds.) 43rd International Colloquium on Automata, Languages and Programming, ICALP 2016. LIPIcs, Rome, Italy, 11–15 July 2016, vol. 55, pp. 31:1–31:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)
Chattopadhyay, E., Li, X.: Explicit non-malleable extractors, multi-source extractors, and almost optimal privacy amplification protocols. In: Dinur, I. (ed.) 57th Annual Symposium on Foundations of Computer Science, New Brunswick, NJ, USA, 9–11 October 2016, pp. 158–167. IEEE Computer Society Press (2016)
Chattopadhyay, E., Zuckerman, D.: Non-malleable codes against constant split-state tampering. In: 55th Annual Symposium on Foundations of Computer Science, Philadelphia, PA, USA, 18–21 October 2014, pp. 306–315. IEEE Computer Society Press (2014)
Cheraghchi, M., Guruswami, V.: Capacity of non-malleable codes. CoRR, abs/1309.0458 (2013)
Cheraghchi, M., Guruswami, V.: Capacity of non-malleable codes. In: Naor, M. (ed.) 5th Innovations in Theoretical Computer Science, ITCS 2014, Princeton, NJ, USA, 12–14 January 2014, pp. 155–168. Association for Computing Machinery (2014)
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
Cramer, R., Dodis, Y., Fehr, S., Padró, C., Wichs, D.: Detection of algebraic manipulation with applications to robust secret sharing and fuzzy extractors. In: Smart, N. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 471–488. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78967-3_27
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)
Dziembowski, S., Kazana, T., Obremski, M.: Non-malleable codes from two-source extractors. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013. LNCS, vol. 8043, pp. 239–257. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40084-1_14
Dziembowski, S., Pietrzak, K., Wichs, D.: Non-malleable codes. In: Yao, A.-C.-C. (ed.) 1st Innovations in Computer Science, ICS 2010, Tsinghua University, Beijing, China, 5–7 January 2010, pp. 434–452. Tsinghua University Press (2010)
Faust, S., Mukherjee, P., Venturi, D., Wichs, D.: Efficient non-malleable codes and key-derivation for poly-size tampering circuits. In: Nguyen, P.Q., Oswald, E. (eds.) EUROCRYPT 2014. LNCS, vol. 8441, pp. 111–128. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-55220-5_7
Goyal, V., Kumar, A.: Non-malleable secret sharing. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, 25–29 June 2018 (2018)
Gupta, D., Maji, H.K., Wang, M.: Non-malleable codes against lookahead tampering. Cryptology ePrint Archive, report 2017/1048 (2017). https://eprint.iacr.org/2017/1048
Guruswami, V., Umans, C., Vadhan, S.P.: Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. In: 22nd Annual IEEE Conference on Computational Complexity (CCC 2007), 13–16 June 2007, San Diego, California, USA, pp. 96–108 (2007)
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. 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. LNCS, vol. 10822, pp. 589–617. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-78372-7_19
Li, X.: Improved non-malleable extractors, non-malleable codes and independent source extractors. In: Hatami, H., McKenzie, P., King, V. (eds.) 49th Annual ACM Symposium on Theory of Computing, Montreal, QC, Canada, 19–23 June 2017, pp. 1144–1156. ACM Press (2017)
Li, X.: Pseudorandom correlation breakers, independence preserving mergers and their applications. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 25, p. 28 (2018)
Vadhan, S.P.: Pseudorandomness. Foundations and Trends in Theoretical Computer Science. Now Publishers (2012)
Author information
Authors and Affiliations
Corresponding authors
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this paper
Cite this paper
Gupta, D., Maji, H.K., Wang, M. (2018). Non-malleable Codes Against Lookahead Tampering. In: Chakraborty, D., Iwata, T. (eds) Progress in Cryptology – INDOCRYPT 2018. INDOCRYPT 2018. Lecture Notes in Computer Science(), vol 11356. Springer, Cham. https://doi.org/10.1007/978-3-030-05378-9_17
Download citation
DOI: https://doi.org/10.1007/978-3-030-05378-9_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-05377-2
Online ISBN: 978-3-030-05378-9
eBook Packages: Computer ScienceComputer Science (R0)