Capacity of Non-Malleable Codes. Cheraghchi, M. & Guruswami, V. IEEE Transactions on Information Theory, 62(3):1097–1118, 2016. Preliminary version in Proceedings of ITCS 2014.Link Paper doi abstract bibtex 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 $\mathcal{F}$ of tampering functions that is not too large (for instance, when $|\mathcal{F}| łe 2^{2^{α n}}$ for some $α <1$ where $n$ is the number of bits in a codeword). In this work, 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. (ICS 2010). Specifically, 1) We prove that for every family $\mathcal{F}$ with $|\mathcal{F}| łe 2^{2^{α n}}$, there exist non-malleable codes against $\mathcal{F}$ with rate arbitrarily close to $1-α$ (this is achieved w.h.p. by a randomized construction). 2) We show the existence of families of size $\exp(n^{O(1)} 2^{α n})$ against which there is no non-malleable code of rate $1-α$ (in fact this is the case w.h.p for a random family of this size). 3) We also show that $1-α$ is the best achievable rate for the family of functions which are only allowed to tamper the first $α n$ 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 $c > 0$ and family $\mathcal{F}$ of size $2^{n^c}$, in particular tampering functions with say cubic size circuits.
@ARTICLE{ref:CG16,
author = {Mahdi Cheraghchi and Venkatesan Guruswami},
title = {Capacity of Non-Malleable Codes},
journal = {IEEE Transactions on Information Theory},
year = 2016,
volume = 62,
number = 3,
pages = {1097--1118},
doi = {10.1109/TIT.2015.2511784},
url_Link = {https://ieeexplore.ieee.org/document/7365445},
keywords = {cryptography, coding theory, tamper-resilient
storage, probabilistic method, information theory,
error detection},
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
$\mathcal{F}$ of tampering functions that is not too
large (for instance, when $|\mathcal{F}| \le
2^{2^{\alpha n}}$ for some $\alpha <1$ where $n$ is
the number of bits in a codeword). In this work, 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. (ICS 2010).
Specifically, 1) We prove that for every family
$\mathcal{F}$ with $|\mathcal{F}| \le 2^{2^{\alpha
n}}$, there exist non-malleable codes against
$\mathcal{F}$ with rate arbitrarily close to
$1-\alpha$ (this is achieved w.h.p. by a randomized
construction). 2) 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). 3) 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 $c
> 0$ and family $\mathcal{F}$ of size $2^{n^c}$, in
particular tampering functions with say cubic size
circuits. },
note = {Preliminary version in Proceedings of {ITCS 2014.}},
url_Paper = {https://eccc.weizmann.ac.il//report/2013/118}
}
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While this is an impossible goal to achieve against unrestricted tampering functions, rather surprisingly non-malleable coding becomes possible against every fixed family $\\mathcal{F}$ of tampering functions that is not too large (for instance, when $|\\mathcal{F}| łe 2^{2^{α n}}$ for some $α <1$ where $n$ is the number of bits in a codeword). In this work, 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. (ICS 2010). Specifically, 1) We prove that for every family $\\mathcal{F}$ with $|\\mathcal{F}| łe 2^{2^{α n}}$, there exist non-malleable codes against $\\mathcal{F}$ with rate arbitrarily close to $1-α$ (this is achieved w.h.p. by a randomized construction). 2) We show the existence of families of size $\\exp(n^{O(1)} 2^{α n})$ against which there is no non-malleable code of rate $1-α$ (in fact this is the case w.h.p for a random family of this size). 3) We also show that $1-α$ is the best achievable rate for the family of functions which are only allowed to tamper the first $α n$ 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 $c > 0$ and family $\\mathcal{F}$ of size $2^{n^c}$, in particular tampering functions with say cubic size circuits. ","note":"Preliminary version in Proceedings of ITCS 2014.","url_paper":"https://eccc.weizmann.ac.il//report/2013/118","bibtex":"@ARTICLE{ref:CG16,\n author =\t {Mahdi Cheraghchi and Venkatesan Guruswami},\n title =\t {Capacity of Non-Malleable Codes},\n journal =\t {IEEE Transactions on Information Theory},\n year =\t 2016,\n volume =\t 62,\n number =\t 3,\n pages =\t {1097--1118},\n doi =\t\t {10.1109/TIT.2015.2511784},\n url_Link =\t {https://ieeexplore.ieee.org/document/7365445},\n keywords =\t {cryptography, coding theory, tamper-resilient\n storage, probabilistic method, information theory,\n error detection},\n abstract =\t {Non-malleable codes, introduced by Dziembowski,\n Pietrzak and Wichs (ICS 2010), encode messages $s$\n in a manner so that tampering the codeword causes\n the decoder to either output $s$ or a message that\n is independent of $s$. While this is an impossible\n goal to achieve against unrestricted tampering\n functions, rather surprisingly non-malleable coding\n becomes possible against every fixed family\n $\\mathcal{F}$ of tampering functions that is not too\n large (for instance, when $|\\mathcal{F}| \\le\n 2^{2^{\\alpha n}}$ for some $\\alpha <1$ where $n$ is\n the number of bits in a codeword). In this work, we\n study the \"capacity of non-malleable codes,\" and\n establish optimal bounds on the achievable rate as a\n function of the family size, answering an open\n problem from Dziembowski et al. (ICS 2010).\n Specifically, 1) We prove that for every family\n $\\mathcal{F}$ with $|\\mathcal{F}| \\le 2^{2^{\\alpha\n n}}$, there exist non-malleable codes against\n $\\mathcal{F}$ with rate arbitrarily close to\n $1-\\alpha$ (this is achieved w.h.p. by a randomized\n construction). 2) We show the existence of families\n of size $\\exp(n^{O(1)} 2^{\\alpha n})$ against which\n there is no non-malleable code of rate $1-\\alpha$\n (in fact this is the case w.h.p for a random family\n of this size). 3) We also show that $1-\\alpha$ is\n the best achievable rate for the family of functions\n which are only allowed to tamper the first $\\alpha\n n$ bits of the codeword, which is of special\n interest. As a corollary, this implies that the\n capacity of non-malleable coding in the split-state\n model (where the tampering function acts\n independently but arbitrarily on the two halves of\n the codeword, a model which has received some\n attention recently) equals $1/2$. We also give an\n efficient Monte Carlo construction of codes of rate\n close to $1$ with polynomial time encoding and\n decoding that is non-malleable against any fixed $c\n > 0$ and family $\\mathcal{F}$ of size $2^{n^c}$, in\n particular tampering functions with say cubic size\n circuits. },\n note =\t {Preliminary version in Proceedings of {ITCS 2014.}},\n url_Paper =\t {https://eccc.weizmann.ac.il//report/2013/118}\n}\n\n","author_short":["Cheraghchi, M.","Guruswami, V."],"key":"ref:CG16","id":"ref:CG16","bibbaseid":"cheraghchi-guruswami-capacityofnonmalleablecodes-2016","role":"author","urls":{" link":"https://ieeexplore.ieee.org/document/7365445"," paper":"https://eccc.weizmann.ac.il//report/2013/118"},"keyword":["cryptography","coding theory","tamper-resilient storage","probabilistic method","information theory","error detection"],"metadata":{"authorlinks":{"cheraghchi, m":"https://mahdi.ch/"}}},"bibtype":"article","biburl":"http://mahdi.ch/writings/cheraghchi.bib","creationDate":"2020-05-28T23:53:05.515Z","downloads":4,"keywords":["cryptography","coding theory","tamper-resilient storage","probabilistic method","information theory","error detection"],"search_terms":["capacity","non","malleable","codes","cheraghchi","guruswami"],"title":"Capacity of Non-Malleable Codes","year":2016,"dataSources":["YZqdBBx6FeYmvQE6D"]}