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TR23-167 | 13th November 2023 16:16

Non-malleable codes with optimal rate for poly-size circuits


Authors: Marshall Ball, Ronen Shaltiel, Jad Silbak
Publication: 13th November 2023 16:16
Downloads: 42


We give an explicit construction of non-malleable codes with rate $1-o(1)$ for the tampering class of poly-size circuits. This rate is optimal, and improves upon the previous explicit construction of Ball, Dachman-Soled and Loss (CRYPTO 2022) which achieves a rate smaller than $\frac{1}{n}$. Our codes are based on the same hardness assumption used by Ball, Dachman-Soled and Loss, namely, that there exists a problem in $\text{E}=\text{DTIME}(2^{O(n)})$ that requires nondeterministic circuits of size $2^{\Omega(n)}$. This is a standard complexity theoretic assumption that was used in many papers in complexity theory and cryptography, and can be viewed as a scaled, nonuniform version of the widely believed assumption that $\text{EXP} \not \subseteq \text{NP}$. Our result is incomparable to that of Ball, Dachman-Soled and Loss, as we only achieve computational (rather than statistical) security. Non-malleable codes with Computational security (with lower error than what we get) were obtained by (Ball et al. 2019, Dachman-Soled, Komargodsky and Pass 2021) under strong cryptographic assumptions. We show that our approach can potentially yield statistical security if certain explicit constructions of pseudorandom objects can be improved.

By composing our new non-malleable codes with standard (information theoretic) error-correcting codes (that recover from a $p$ fraction of errors) we achieve the \emph{best of both worlds}. Namely, we achieve explicit codes that recover from a $p$-fraction of errors and have the same rate as the best known explicit information theoretic codes, while \emph{also} being non-malleable for poly-size circuits.

Moreover, if we restrict our attention to errors that are introduced by poly-size circuits, we can achieve best of both worlds codes with rate $1-H(p)$. This is superior to the rate achieved by standard (information theoretic) error-correcting codes, and this result is obtained by composing our new non-malleable codes with the recent codes of Shaltiel and Silbak (ECCC 2023).

Our technique combines ideas from non-malleable codes and pseudorandomness. We show how to take a low rate ``small set non-malleable code (this is a variant of non-malleable codes with a different notion of security that was introduced by Shaltiel and Silbak (FOCS 2022) and compile it into a (standard) high-rate non-malleable code. Using small set non-malleable codes (as well as seed-extending PRGs) bypasses difficulties that arise when analysing standard non-malleable codes, and allows us to use a simple construction.

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