Non-malleable codes, introduced by Dziembowski, Pietrzak, and Wichs (ICS ’10) provide the guarantee that if a codeword c of a message m, is modified by a tampering function f to c’, then c’ either decodes to m or to “something unrelated” to m. It is known that non-malleable codes cannot exist for the class of all tampering functions and hence a lot of work has focused on explicitly constructing such codes against a large and natural class of tampering functions. One such popular, but restricted, class is the so-called split-state model in which the tampering function operates on different parts of the codeword independently. In this work, we consider a stronger adversarial model called block-wise tampering model, in which we allow tampering to depend on more than one block: if a codeword consists of two blocks c = (c1; c2), then the first tampering function f1 could produce a tampered part c’1 = f1(c1) and the second tampering function f2 could produce c’2 = f2(c1; c2) depending on both c2 and c1. The notion similarly extends to multiple blocks where tampering of block ci could happen with the knowledge of all cj for j<=i. We argue this is a natural notion where, for example, the blocks are sent one by one and the adversary must send the tampered block before it gets the next block. A little thought reveals however that one cannot construct such codes that are non-malleable (in the standard sense) against such a powerful adversary: indeed, upon receiving the last block, an adversary could decode the entire codeword and then can tamper depending on the message.<\/p>\n
In light of this impossibility, we consider a natural relaxation called non-malleable codes with replacement which requires the adversary to produce not only related but also a valid codeword in order to succeed. Unfortunately, we show that even this relaxed definition is not achievable in the information-theoretic setting (i.e., when the tampering functions can be unbounded) which implies that we must turn our attention towards computationally bounded adversaries.<\/p>\n
As our main result, we show how to construct block-wise non-malleable codes from sub-exponentially hard one-way permutations. Moreover, we provide an interesting connection between block-wise non-malleable codes and non-malleable commitments. We show that any block-wise nonmalleable code can be converted into a non-malleable (w.r.t. opening) commitment scheme. Our techniques, quite surprisingly, give rise to a non-malleable commitment scheme (secure against so-called synchronizing adversaries), in which only the committer sends messages. We believe this result to be of independent interest. In the other direction, we show that any non-interactive non-malleable (w.r.t. opening) commitment can be used to construct a block-wise non-malleable code only with 2 blocks. Unfortunately, such commitment scheme exists only under highly non-standard assumptions (adaptive one-way functions) and hence can not substitute our main construction.<\/p>\n","protected":false},"excerpt":{"rendered":"
Non-malleable codes, introduced by Dziembowski, Pietrzak, and Wichs (ICS ’10) provide the guarantee that if a codeword c of a message m, is modified by a tampering function f to c’, then c’ either decodes to m or to “something unrelated” to m. 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