Non-malleable codes from additive combinatorics

Non-malleable codes provide a useful and meaningful security guarantee in situations where traditional error-correction (and even error-detection) is impossible, for example, when the attacker can completely overwrite the encoded message. Informally, a code is non-malleable if the message contained in a modified codeword is either the original message or a completely unrelated value. Although such codes do not exist if the family of “tampering functions” F is completely unrestricted, they are known to exist for many broad tampering families F. One such natural family is the family of tampering functions in the so-called split-state model. Here the message m is encoded into two shares L and R, and the attacker is allowed to arbitrarily tamper with L and R individually. The split-state tampering arises in many realistic applications, such as the design of non-malleable secret sharing schemes, motivating the question of designing efficient non-malleable codes in this model. Prior to this work, non-malleable codes in the split-state model received considerable attention in the literature but either (1) were constructed in the random oracle model, or (2) relied on advanced cryptographic assumptions (such as noninteractive zero-knowledge proofs and leakage-resilient encryption), or (3) could only encode 1-bit messages. As our main result, we build the first efficient, multi-bit, information-theoretically-secure non-malleable code in the split-state model. The heart of our construction uses the following new property of the inner-product function L, R over the vector space Fpⁿ (for a prime p and large enough dimension n): if L and R are uniformly random over Fpⁿ, and f, g : Fpⁿ → Fpⁿ are two arbitrary functions on L and R, then the joint distribution (L, R, f(L), g(R)) is “close” to the convex combination of “affine distributions” ((U, aU + b) | a, b ∈ Fp), where U is uniformly random in Fp. In turn, the proof of this surprising property of the inner product function critically relies on some results from additive combinatorics, including the so-called quasi-polynomial Freiman–Ruzsa theorem, which was recently established by Sanders [Anal. PDE, 5 (2012), pp. 627–655] as a step toward resolving the polynomial Freiman–Ruzsa conjecture [B. Green, in Surveys in Combinatorics, London Mathematical Society, London, 2005, pp. 1–29].