Linear-Time Non-Malleable Codes in the Bit-Wise Independent Tampering Model

Abstract

Abstract. Non-malleable codes were introduced by Dziembowski et al. (ICS 2010) as coding schemes that protect a message against tampering attacks. Roughly speaking, a code is non-malleable if decoding an adversarially tampered encoding of a message m produces the original message m or a value m (eventually ⊥) completely unrelated with m. It is known that non-malleability is possible only for restricted classes of tampering functions. Since their introduction, a long line of works has established feasibility results of non-malleable codes against different families of tampering functions. However, for many interesting families the challenge of finding "good" non-malleable codes remains open. In particular, we would like to have explicit constructions of non-malleable codes with high-rate and efficient encoding/decoding algorithms (i.e. low computational complexity). In this work we present two explicit constructions: the first one is a natural generalization of the work of Dziembowski et al. and gives rise to the first constant-rate non-malleable code with linear-time complexity (in a model including bit-wise independent tampering). The second construction is inspired by the recent works about non-malleable codes of Agrawal et al. (TCC 2015) and of Cheraghchi and Guruswami (TCC 2014) and improves the previous result in the bit-wise tampering model: it builds the first non-malleable codes with linear-time complexity and optimal-rate (i.e. rate 1 − o(1)).

Coding in theoretical computer science

“…But their codes can not run in linear time. In [20], they present a linear-time non-malleable code resistant to bit-wise tampering functions. In Chapter 6, we follow their works and provide a linear-time non-malleable code resistant to both bit-wise and permutation tampering functions.…”

“…In [20], they first construct a constant rate non-malleable code. This non-malleable code is not necessarily linear-time.…”

“…In this chapter, we construct a linear-time non-malleable code with respect to bit-wise tampering and permutation functions. The key idea is combining the construction in [4] and [20].…”

“…Thus, we take it as a short code. The construction in [20] is our long code. Although this idea seems natural and feasible, we take much effort to prove its correctness.…”

“…The statistical distance of two random variables X and Y with finite range S is defined by To construct our linear-time non-malleable codes, we still need the linear-time hash function family. We use the construction in [20] as our building blocks. Here, we assume that r is a divisor of n. Let CM i = { (i−1)n r + j : j = 0, .…”