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Capacity of Non-Malleable Codes.
- Source :
-
IEEE Transactions on Information Theory . Mar2016, Vol. 62 Issue 3, p1097-1118. 22p. - Publication Year :
- 2016
-
Abstract
- Non-malleable codes, introduced by Dziembowski et al., 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 of tampering functions that is not too large (for instance, when | \mathcal {F}| \leqslant 2^{2^{\vphantom {R^{a}}\alpha n}} for some \alpha <1 , where n is the number of bits in a codeword). In this paper, we study the capacity of non-malleable codes, and establish optimal bounds on the achievable rate as a function of the family size, answering an open problem from Dziembowski et al. Specifically, We prove that for every family \mathcal {F} with | \mathcal {F}| \leqslant 2^{2^{\vphantom {R^{}}\alpha n}} , there exist non-malleable codes against \mathcal F with rate arbitrarily close to 1-\alpha [this is achieved with high probability (w.h.p.) by a randomized construction]. We show the existence of families of size \exp (n^O(1) 2^\alpha n) against which there is no non-malleable code of rate $1-\alpha $ (in fact this is the case w.h.p for a random family of this size). We also show that 1-\alpha $ is the best achievable rate for the family of functions, which are only allowed to tamper the first \alpha n$ 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, a model which has received some attention recently) equals 1/2. 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 c > 0 of size 2^{n^{c}} , in particular tampering functions with, say, cubic size circuits. [ABSTRACT FROM PUBLISHER]
Details
- Language :
- English
- ISSN :
- 00189448
- Volume :
- 62
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- IEEE Transactions on Information Theory
- Publication Type :
- Academic Journal
- Accession number :
- 113114889
- Full Text :
- https://doi.org/10.1109/TIT.2015.2511784