Non-malleable codes and extractors for small-depth circuits, and affine functions

Abstract

Non-malleable codes were introduced by Dziembowski, Pietrzak and Wichs as an elegant relaxation of error correcting codes, where the motivation is to handle more general forms of tampering while still providing meaningful guarantees. This has led to many elegant constructions and applications in cryptography. However, most works so far only studied tampering in the split-state model where di erent parts of the codeword are tampered independently, and thus do not apply to many other natural classes of tampering functions. The only exceptions are the work of Agrawal et al, which studied non-malleable codes against bit permutation composed with bit-wise tampering, and the works of Faust et al and Ball et al, which studied non-malleable codes against local functions. However, in both cases each tampered bit only depends on a subset of input bits.In this work, we study the problem of constructing non-malleable codes against more general tampering functions that act on the entire codeword. We give the rst e cient constructions of nonmalleable codes against AC 0 tampering functions and a ne tampering functions. These are the rst explicit non-malleable codes against tampering functions where each tampered bit can depend on all input bits. We also give e cient non-malleable codes against t-local functions for t = o( √ n), where a t-local function has the property that any output bit depends on at most t input bits. In the case of deterministic decoders, this improves upon the results of Ball et al, which can handle t ≤ n 1 4 . All our results on non-malleable codes are obtained by using the connection between non-malleable codes and seedless nonmalleable extractors discovered by Cheraghchi and Guruswami. Therefore, we also give the rst e cient constructions of seedless non-malleable extractors against AC 0 tampering functions, t-local tampering functions for t = o( √ n), and a ne tampering functions.To derive our results on non-malleable codes, we design e cient algorithms to almost uniformly sample from the pre-image of any given output of our non-malleable extractor.

Non-Malleable Codes for Small-Depth Circuits

“…In this work, we address the main open problem from [CL17]: we give the first explicit construction of non-malleable codes for small-depth circuits achieving polynomial rate:…”

“…[ADKO15] showed that a non-malleable code for the simpler G, when concatenated with an (inner) non-malleable reduction (E, D) from F to G, yields a non-malleable code for the more "complex" F. (See Remark 4 for a comparison of our approach to that of [CL17]. )…”

“…Remark 4 (Relation to the techniques of [CL17].). Although Chattopadhyay and Li [CL17] also use the switching lemma in their work, our overall approach is essentially orthogonal to theirs. At a high level, [CL17] uses a framework of Cheraghchi and Guruswami [CG16] to derive nonmalleable codes from non-malleable extractors.…”

“…In this framework, the rate of the code is directly tied to the error of the extractor; roughly speaking, as the parameters of the switching lemma can be at best inverse-quasipolynomial when reducing to local functions, this unfortunately translates (via the [CG16] framework) into codes with at best exponentially small rate (see pg. 10 of [CL17] for a discussion of this issue). Circumventing this limitation therefore necessitates a significantly different approach, and indeed, as discussed above we construct our non-malleable codes without using extractors as an intermediary.…”

Non-malleable Coding for Arbitrary Varying Channels

“…We will also use non-malleable codes for the family of affine tampering functions f : {0, 1} n → {0, 1} n , where each output bit of a tampering function can be written as an affine function of the n input bits. Non-malleable codes for this family were explicitly constructed in [3]. Our work.…”