# Non-malleable Coding Against Bit-Wise and Split-State Tampering

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## Abstract

Non-malleable coding, introduced by Dziembowski et al. (ICS 2010), aims for protecting the integrity of information against tampering attacks in situations where error detection is impossible. Intuitively, information encoded by a non-malleable code either decodes to the original message or, in presence of any tampering, to an unrelated message. Non-malleable coding is possible against any class of adversaries of bounded size. In particular, Dziembowski et al. show that such codes exist and may achieve positive rates for any class of tampering functions of size at most \(2^{2^{\alpha n}}\), for any constant \(\alpha \in [0, 1)\). However, this result is existential and has thus attracted a great deal of subsequent research on explicit constructions of non-malleable codes against natural classes of adversaries. In this work, we consider constructions of coding schemes against two well-studied classes of tampering functions; namely, bit-wise tampering functions (where the adversary tampers each bit of the encoding independently) and the much more general class of split-state adversaries (where two independent adversaries arbitrarily tamper each half of the encoded sequence). We obtain the following results for these models. (1) For bit-tampering adversaries, we obtain explicit and efficiently encodable and decodable non-malleable codes of length *n* achieving rate \(1-o(1)\) and error (also known as “exact security”) \(\exp (-\tilde{\varOmega }(n^{1/7}))\). Alternatively, it is possible to improve the error to \(\exp (-\tilde{\varOmega }(n))\) at the cost of making the construction Monte Carlo with success probability \(1-\exp (-\varOmega (n))\) (while still allowing a compact description of the code). Previously, the best known construction of bit-tampering coding schemes was due to Dziembowski et al. (ICS 2010), which is a Monte Carlo construction achieving rate close to .1887. (2) We initiate the study of *seedless non-malleable extractors* as a natural variation of the notion of non-malleable extractors introduced by Dodis and Wichs (STOC 2009). We show that construction of non-malleable codes for the split-state model reduces to construction of non-malleable two-source extractors. We prove a general result on existence of seedless non-malleable extractors, which implies that codes obtained from our reduction can achieve rates arbitrarily close to 1 / 5 and exponentially small error. In a separate recent work, the authors show that the optimal rate in this model is 1 / 2. Currently, the best known explicit construction of split-state coding schemes is due to Aggarwal, Dodis and Lovett (ECCC TR13-081) which only achieves vanishing (polynomially small) rate.

## Keywords

Information theory Tamper-resilient cryptography Coding theory Error detection Randomness extractors## Notes

### Acknowledgments

The authors would like to thank anonymous referees for their careful reading of an earlier draft of this work and their numerous helpful comments.

## References

- 1.D. Aggarwal, Y. Dodis, T. Kazana, M. Obremski, Non-malleable reductions and applications, in
*Cryptology*ePrint Archive, Report 2014/821 (2014). http://eprint.iacr.org/ - 2.D. Aggarwal, Y. Dodis, S. Lovett, Non-malleable codes from additive combinatorics, in
*Proceedings of the 46th Annual ACM Symposium on Theory of Computing*(2014), pp.774–783Google Scholar - 3.B. Barak, A. Rao, R. Shaltiel, A. Wigderson, 2-Source dispersers for sub-polynomial entropy and Ramsey graphs beating the Frankl–Wilson construction.
*Ann. Math.***176**(3), 1483–1544 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 4.J. Bourgain, More on the Sum–Product phenomenon in prime fields and its applications.
*Int. J. Number Theory***1**(1), 1–32 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 5.E. Chattopadhyay, V. Goyal, X. Li, Non-malleable extractors and codes, with their many tampered extensions. Preprint arXiv:1505.00107 (2015)
- 6.E. Chattopadhyay, D. Zuckerman, Non-malleable codes against constant split-state tampering, in
*Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS)*(2014), pp. 306–315Google Scholar - 7.M. Cheraghchi,
*Applications of Derandomization Theory in Coding*. Ph.D. Thesis, Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland (2010). http://eccc.hpi-web.de/static/books/Applications_of_Derandomization_Theory_in_Coding/ - 8.M. Cheraghchi, V. Guruswami, Capacity of non-malleable codes, in
*Proceedings of Innovations in Theoretical Computer Science (ITCS 2014)*(2014)Google Scholar - 9.M. Cheraghchi, V. Guruswami, Non-malleable coding against bit-wise and split-state tampering, in
*Proceedings of Theory of Cryptography Conference (TCC 2014)*(2014)Google Scholar - 10.B. Chor, O. Goldreich, Unbiased bits from sources of weak randomness and probabilistic communication complexity.
*SIAM J. Comput.*,**2**(17), 230–261 (1988)MathSciNetCrossRefzbMATHGoogle Scholar - 11.R. Cramer, H. Chen, S. Goldwasser, R. de Haan, V. Vaikuntanathan, Secure computation from random error-correcting codes, in
*Proceedings of Eurocrypt 2007*(2007), pp. 291–310Google Scholar - 12.R. Cramer, Y. Dodis, S. Fehr, C. Padró, D. Wichs, Detection of algebraic manipulation with applications to robust secret sharing and fuzzy extractors, in
*Proceedings of EUROCRYPT 2008*(2008), pp. 471–488Google Scholar - 13.Y. Dodis, D. Wichs, Non-malleable extractors and symmetric key cryptography from weak secrets, in
*Proceedings of the 41st annual ACM Symposium on Theory of Computing*(2009), pp. 601–610. Full version published in Cryptology ePrint Archive, Report 2008/503 (eprint.iacr.org/2008/503)Google Scholar - 14.S. Dziembowski, T. Kazana, M. Obremski, Non-malleable codes from two-source extractors, in
*Proceedings of CRYPTO*(2013), pp. 239–257Google Scholar - 15.S. Dziembowski, K. Pietrzak, D. Wichs, Non-malleable codes, in
*Proceedings of Innovations in Computer Science (ICS 2010)*(2010)Google Scholar - 16.G.D. Forney,
*Concatenated Codes*(MIT Press, Cambridge, 1966)zbMATHGoogle Scholar - 17.V. Guruswami, A. Smith. Codes for computationally simple channels: Explicit constructions with optimal rate, in
*Proceedings of FOCS 2010*(2010), pp. 723–732Google Scholar - 18.J. Justesen, A class of constructive asymptotically good algebraic codes.
*IEEE Trans. Inf. Theory***18**, 652–656 (1972)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Y. Kalai, X. Li, A. Rao, in
*2th Annual IEEE Symposium on Foundations of Computer Science (FOCS)*(2009), pp. 617–626Google Scholar - 20.E. Kaplan, M. Naor, O. Reingold, Derandomized constructions of \(k\)-wise (almost) independent permutations, in
*Proceedings of RANDOM 2005*(2005), pp. 113–133Google Scholar - 21.A. Rao, A 2-source almost-extractor for linear entropy, in
*Proceedings of RANDOM 2008*(2008), pp. 549–556Google Scholar - 22.R. Raz, Extractors with weak random seeds, in
*Proceedings of the37th Annual ACM Symposium on Theory of Computing (STOC)*(2005), pp. 11–20Google Scholar - 23.R. Raz, A. Yehudayoff, Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors.
*J. Comput. Syst. Sci***77**(1), 167–190 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 24.S. Vadhan, Pseudorandomness.
*Found. Trends Theor. Comput. Sci.***7**(1–3), 1–336 (2012)MathSciNetCrossRefzbMATHGoogle Scholar