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Four-State Non-malleable Codes with Explicit Constant Rate

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Abstract

Non-malleable codes (NMCs), introduced by Dziembowski, Pietrzak and Wichs (ITCS 2010), provide a powerful guarantee in scenarios where the classical notion of error-correcting codes cannot provide any guarantee: a decoded message is either the same or completely independent of the underlying message, regardless of the number of errors introduced into the codeword. Informally, NMCs are defined with respect to a family of tampering functions \(\mathcal {F}\) and guarantee that any tampered codeword decodes either to the same message or to an independent message, so long as it is tampered using a function \(f \in \mathcal {F}\). One of the well-studied tampering families for NMCs is the t-split-state family, where the adversary tampers each of the t“states” of a codeword, arbitrarily but independently. Cheraghchi and Guruswami (TCC 2014) obtain a rate-1 non-malleable code for the case where \(t = \mathcal {O}(n)\) with n being the codeword length and, in (ITCS 2014), show an upper bound of \(1-1/t\) on the best achievable rate for any t-split state NMC. For \(t=10\), Chattopadhyay and Zuckerman (FOCS 2014) achieve a constant-rate construction where the constant is unknown. In summary, there is no known construction of an NMC with an explicit constant rate for any \(t= o(n)\), let alone one that comes close to matching Cheraghchi and Guruswami’s lowerbound! In this work, we construct an efficient non-malleable code in the t-split-state model, for \(t=4\), that achieves a constant rate of \(\frac{1}{3+\zeta }\), for any constant \(\zeta > 0\), and error \(2^{-\varOmega (\ell / log^{c+1} \ell )}\), where \(\ell \) is the length of the message and \(c > 0\) is a constant.

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Notes

  1. For example, this input–output behaviour may be decryption of ciphertexts in the case of Chosen Ciphertext Security of Encryption or signatures of messages in the case of Digital Signatures.

  2. LECSS ensures that the bits of a codeword are t-wise independent and detects tampering if the codeword is modified by an offset \(\Delta \), when \(\Delta \) is not a valid codeword of the scheme.

  3. AMD codes detect tampering attacks that add some pre-determined offset \(\Delta \) to the codeword.

  4. This tampering family captures other tampering attacks such as bit-wise tampering, identity function, and constant function. A motivation to study this model comes from practical applications like cloud storage, where a single file may be stored in t parts at t different locations and an adversary tampers each of these parts independent of the other. It is therefore both of theoretical and practical interest to obtain non-malleable codes for the t-split-state family where \(t>1\) is as small as possible.

  5. Specifically, Liu and Lysyanskaya [34] present a computational non-malleable code w.r.t. split-state tampering functions in the common reference string (CRS) model, using number theoretic assumptions and assuming existence of robust non-interactive zero-knowledge proof systems for an appropriate NP language.

  6. This problem does not arise with a \(\textsf {MAC}\) such as \(ax+b\) where (a, b) is the \(\textsf {MAC}\) key and x is the underlying message. There, for a fixed key and fixed tag, there is a unique message which satisfies the linear equation.

  7. We ensure this by encoding \(s\) using a non-malleable code.

  8. It is crucial to authenticate them separately as, a construction where we do not authenticate them separately is insecure. This is brought out in the security proof later.

  9. A key point to note is that the existing efficient 2-state non-malleable code constructions [3, 32] are augmented-non-malleable codes, and hence we can in fact start with a 2-split-state augmented-non-malleable code.

  10. Although the paper explicitly doesn’t state that the construction is augmented, the construction is in fact an augmented-non-malleable code(as observed in [31]).

  11. The explicit two-state non-malleable code construction given in [32] is in fact an augmented-non-malleable code (as observed in [31]).

  12. The preliminary version of this paper at TCC 2017 did not show the augmented-non-malleability feature.

  13. Capital letters \(L_1, C\) and \({\tilde{M}}\) denote the distributions on respective states and the tampered message.

  14. If we instantiate our construction with the improved rate (\(\varOmega ((\log \log \log n)/(\log \log n))\)) construction of [33], we get an improved error, but our rate remains the same.

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Acknowledgements

We thank Yevgeniy Dodis for insightful comments related to the generalization in Sect. 5. We also thank the anonymous referees for several helpful comments. Research of the first author was supported, in part, by Department of Science and Technology Inspire Faculty Award.

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Authors and Affiliations

  1. Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India

    Bhavana Kanukurthi & Sai Lakshmi Bhavana Obbattu

  2. Department of Mathematics, Indian Institute of Science, Bangalore, India

    Sruthi Sekar

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  1. Bhavana Kanukurthi
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  2. Sai Lakshmi Bhavana Obbattu
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Correspondence to Sruthi Sekar.

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A preliminary version of this paper appeared in the Proceedings of TCC 2017.

A Appendix

A Appendix

Lemma 9

If \( \alpha = \varOmega (\dfrac{\beta }{\log (\beta )}) \), then \( \beta = \mathcal {O}(\alpha .\log (\alpha )) \)

Proof

By the definition of \( \varOmega \), \( \exists \) a constant \( c>0 \) such that for large \( \alpha , \beta \)

$$\begin{aligned}&0\le c.\dfrac{\beta }{\log (\beta )} \le \alpha \nonumber \\&c \beta \le \alpha . \log (\beta ) \nonumber \\&c \beta \le \alpha \sqrt{\beta } \end{aligned}$$
(15)

If \( c\ge 1 \)

$$\begin{aligned}&\sqrt{\beta } \le \alpha \\&\log (\beta ) \le 2.\log (\alpha ) \end{aligned}$$

Multiplying with Eq. 15, we get

$$\begin{aligned} 0 \le \dfrac{c}{2}. \beta \le \alpha \log (\alpha ) \end{aligned}$$
(16)

If \( c<1 \), let \( c' = \dfrac{1}{c} \)

$$\begin{aligned}&\sqrt{\beta } \le c' .\alpha \\&\log (\beta ) \le 2(\log (c') + \log (\alpha )) \\&\log (\beta ) \le 4.\log (\alpha ) \end{aligned}$$

Multiplying with Eq. 15

$$\begin{aligned} 0 \le \dfrac{c}{4}. \beta \le \alpha \log (\alpha ) \end{aligned}$$
(17)

In either case, for large \( \alpha , \beta \), for a constant \( \dfrac{c}{4}>0 \)

$$\begin{aligned}&0 \le \dfrac{c}{4}. \beta \le \alpha \log (\alpha )\\&\quad \implies \alpha \log (\alpha ) = \varOmega (\beta ) \\&\quad \implies \beta = \mathcal {O}(\alpha \log (\alpha )) \end{aligned}$$

\(\square \)

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Kanukurthi, B., Obbattu, S.L.B. & Sekar, S. Four-State Non-malleable Codes with Explicit Constant Rate. J Cryptol 33, 1044–1079 (2020). https://doi.org/10.1007/s00145-019-09339-7

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