There are natural cryptographic applications where an adversary only gets to tamper a high-speed data stream on the fly based on her view so far, namely, the lookahead tampering model. Since the adversary can easily substitute transmitted messages with her messages, it is farfetched to insist on strong guarantees like error-correction or, even, manipulation detection. Dziembowski, Pietrzak, and Wichs (ICS-2010) introduced the notion of non-malleable codes that provide a useful message integrity for such scenarios. Intuitively, a non-malleable code ensures that the tampered codeword encodes the original message or a message that is entirely independent of the original message. Our work studies the following tampering model. We encode a message into k ≥ 1 secret shares, and we transmit each share as a separate stream of data. Adversaries can perform lookahead tampering on each share, albeit, independently. We call this k-lookahead model. First, we show a hardness result for the k-lookahead model. To transmit an ℓ-bit message, the cumulative length of the secret shares must be at least k/k-1ℓ. This result immediately rules out the possibility of a solution with k = 1. Next, we construct a solution for 2-lookahead model such that the total length of the shares is 3ℓ, which is only 1.5x of the optimal encoding as indicated by our hardness result. Prior work considers stronger model of split-state encoding that creates k ≥ 2 secret shares, but protects against adversaries who perform arbitrary (but independent) tampering on each secret share. The size of the secret shares of the most efficient 2-split-state encoding is ℓlog ℓ/log logℓ (Li, ECCC-2018). Even though k-lookahead is a weaker tampering class, our hardness result matches that of k-split-state tampering by Cheraghchi and Guruswami (TCC-2014). However, our explicit constructions above achieve much higher efficiency in encoding.
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