Capacity of Non-Malleable Codes
Capacity of Non-Malleable Codes
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不可延展代码的容量
DOI：
10.1145/2554797.2554814
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发表时间：
2013
影响因子：
2.5
通讯作者：
V. Guruswami
中科院分区：
文献类型：
--
作者：
Mahdi Cheraghchi;V. Guruswami
由Dziembowski等人引入的非易纪录代码以某种方式编码消息，因此篡改代码字会导致解码器的输出S或与S独立的消息无限制。篡改函数，对于不太大的篡改函数的每个固定族P，当I≤i22αn对于某些α<; 1，1时其中n是代码字中的位数）。具体而言，我们证明，对于IFII≤i22αN的每个家庭，都存在不可易于的代码，而P速率是任意接近1-α的[这是通过A随机结构]。 。 - 在拆分状态中进行编码模型（篡改函数独立起作用，但在密码字的两个半部分中独立起作用，该模型最近受到了一些关注）等于1/2。对于任何固定的C> 0和大小2NC的家族P，尤其是用立方大小的电路篡改功能，对任何固定的C> 0和家族P不可夸大。
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 P of tampering functions that is not too large (for instance, when I≤I 22αn for some α <; 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 P with IFI I≤I 22αn, there exist non-malleable codes against P with rate arbitrarily close to 1-α [this is achieved with high probability (w.h.p.) by a randomized construction]. We show the existence of families of size exp(nO(1)2αn) 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). We also show that 1 - α is the best achievable rate for the family of functions, which are only allowed to tamper the first αn 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 P of size 2nc, in particular tampering functions with, say, cubic size circuits.