Explicit Non-Malleable Codes Resistant to Permutations

The notion of non-malleable codes was introduced as a relaxation of standard error-correction and error-detection. Informally, a code is non-malleable if the message contained in a modified codeword is either the original message, or a completely unrelated value. In the information theoretic setting, although existence of such codes for various rich classes of tampering functions is known, explicit constructions exist only for highly structured family of tampering functions. Prior explicit constructions of non-malleable codes rely on the “compartmentalized” structure of the tampering function, i.e. the codeword is partitioned into a priori fixed blocks and each block can only be tampered independently. The prominent examples of this model are the family of bit-wise independent tampering functions and the split-state model. We consider an infinitely large natural class of non-compartmentalized tampering functions. In our model, the tampering function can permute the bits of the encoding and (optionally) perturb them. In the information theoretic setting, we provide an explicit and efficient, rate-1 non-malleable code for multi-bit messages. Lack of explicit constructions of non-malleable codes for non-compartmentalized tampering functions severely inhibits their utility in cryptographic protocols. As a motivation for our construction, we show an application of non-malleable codes to cryptographic protocols. In an idealized setting, we show how string commitments can be based on one-bit commitments, if non-malleable codes exist. Further, as an example of a non-trivial use of non-malleable codes in standard cryptographic protocols (not in an idealized model), we show that if explicit nonmalleable codes are obtained for a slightly larger class of tampering functions than we currently handle, one can obtain a very simple non-malleable commitment scheme, under somewhat strong assumptions.