Four-State Non-malleable Codes with Explicit Constant Rate

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.

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 $\dfrac{c}{4}>0$

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

$\square$