International Association
for Cryptologic Research

Can Alice and Bob agree on a uniformly random secret key without having any truly secret randomness to begin with? Here we consider a setting where Eve can get partial leakage on the internal state of both Alice and Bob individually before the protocol starts. They then run a protocol using their states without any additional randomness and need to agree on a shared key that looks uniform to Eve, even after observing the leakage and the protocol transcript. We focus on non-interactive (one round) key exchange (NIKE), where Alice and Bob send one message each without waiting for one another. We first consider this problem in the symmetric-key setting, where the states of Alice and Bob include a shared secret as well as individual uniform randomness. However, since Eve gets leakage on these states, Alice and Bob need to perform privacy amplification to derive a fresh secret key from them. Prior solutions require Alice and Bob to sample fresh uniform randomness during the protocol, while in our setting all of their randomness was already part of their individual states a priori and was therefore subject to leakage. We show an information-theoretic solution to this problem using a novel primitive that we call a two-seed extractor, which we in turn construct by drawing a connection to communication-complexity lower-bounds in the number-on-forehead (NOF) model. We then turn to studying this problem in the public-key setting, where the states of Alice and Bob consist of independent uniform randomness. Unfortunately, we give a black-box separation showing that leakage-resilient NIKE in this setting cannot be proven secure via a black-box reduction under any game-based assumption when the leakage is super-logarithmic. This includes virtually all assumptions used in cryptography, and even very strong assumptions such as indistinguishability obfuscation (iO). Nevertheless, we also provide positive results that get around the above separation: -We show that every key exchange protocol (e.g., Diffie-Hellman) is secure when the leakage amount is logarithmic, or potentially even greater if we assume sub-exponential security without leakage. -We notice that the black-box separation does not extend to schemes in the common reference string (CRS) model, or to schemes with preprocessing, where Alice and Bob can individually pre-process their random coins to derive their secret state prior to leakage. We give a solution in the CRS model with preprocessing using bilinear maps. We also give solutions in just the CRS model alone (without preprocessing) or just with preprocessing (without a CRS), using iO and lossy functions.

Non-malleable codes were introduced by Dziembowski, Pietrzak, and Wichs (JACM 2018) as a generalization of standard error correcting codes to handle severe forms of tampering on codewords. This notion has attracted a lot of recent research, resulting in various explicit constructions, which have found applications in tamper-resilient cryptography and connections to other pseudorandom objects in theoretical computer science. We continue the line of investigation on explicit constructions of non-malleable codes in the information theoretic setting, and give explicit constructions for several new classes of tampering functions. These classes strictly generalize several previously studied classes of tampering functions, and in particular extend the well studied split-state model which is a "compartmentalized" model in the sense that the codeword is partitioned \emph{a prior} into disjoint intervals for tampering. Specifically, we give explicit non-malleable codes for the following classes of tampering functions. (1) Interleaved split-state tampering: Here the codeword is partitioned in an unknown way by an adversary, and then tampered with by a split-state tampering function. (2) Affine tampering composed with split-state tampering: In this model, the codeword is first tampered with by a split-state adversary, and then the whole tampered codeword is further tampered with by an affine function. In fact our results are stronger, and we can handle affine tampering composed with interleaved split-state tampering. Our results are the first explicit constructions of non-malleable codes in any of these tampering models. As applications, they also directly give non-malleable secret-sharing schemes with binary shares in the split-state joint tampering model and the stronger model of affine tampering composed with split-state joint tampering. We derive all these results from explicit constructions of seedless non-malleable extractors, which we believe are of independent interest. Using our techniques, we also give an improved seedless extractor for an unknown interleaving of two independent sources.