Paper 2013/564

Capacity of Non-Malleable Codes

Mahdi Cheraghchi and Venkatesan Guruswami

Abstract

Non-malleable codes, introduced by Dziembowski, Pietrzak and Wichs (ICS 2010), encode messages $s$ in a manner so that tampering the codeword causes the decoder to either output $s$ or a message that is independent of $s$. While this is an impossible goal to achieve against unrestricted tampering functions, rather surprisingly non-malleable coding becomes possible against every fixed family $F$ of tampering functions that is not too large (for instance, when $|F|\le \exp (2^{\alpha n})$ for some $\alpha \in [0,1)$ where $n$ is the number of bits in a codeword). In this work, we study the "capacity of non-malleable coding", and establish optimal bounds on the achievable rate as a function of the family size, answering an open problem from Dziembowski et al. (ICS 2010). Specifically, 1. We prove that for every family $$ with $$, there exist non-malleable codes against $$ with rate arbitrarily close to $$ (this is achieved w.h.p. by a randomized construction). 2. We show the existence of families of size $$ against which there is no non-malleable code of rate $$ (in fact this is the case w.h.p for a random family of this size). 3. We also show that $$ is the best achievable rate for the family of functions which are only allowed to tamper the first $$ bits of the codeword, which is of special interest. As a corollary, this implies that the capacity of non-malleable coding in the split-state model (where the tampering function acts independently but arbitrarily on the two halves of the codeword) equals $$. We also give an efficient Monte Carlo construction of codes of rate close to 1 with polynomial time encoding and decoding that is non-malleable against any fixed $$ and family $$ of size $$, in particular tampering functions with, say, cubic size circuits.