Incompressible encodings

An incompressible encoding can probabilistically encode some data m into a codeword c, which is not much larger. Anyone can decode the codeword c to recover the original data m. However, the codeword c cannot be efficiently compressed, even if the original data m is given to the decompression procedure on the side. In other words, c is an efficiently decodable representation of m, yet is computationally incompressible even given m. An incompressible encoding is composable if many encodings cannot be simultaneously compressed.The recent work of Damgard, Ganesh and Orlandi (CRYPTO '19) defined a variant of incompressible encodings as a building block for "proofs of replicated storage". They constructed incompressible encodings in an ideal permutation model, but it was left open if they can be constructed under standard assumptions, or even in the more basic random-oracle model. In this work, we undertake the comprehensive study of incompressible encodings as a primitive of independent interest and give new constructions, negative results and applications:We construct incompressible encodings in the common random string (CRS) model under either Decisional Composite Residuosity (DCR) or Learning with Errors (LWE). However, the construction has several drawbacks: (1) it is not composable, (2) it only achieves selective security, and (3) the CRS is as long as the data m.We leverage the above construction to also get a scheme in the random-oracle model, under the same assumptions, that avoids all of the above drawbacks. Furthermore, it is significantly more efficient than the prior ideal-model construction.We give black-box separations, showing that incompressible encodings in the plain model cannot be proven secure under any standard hardness assumption, and incompressible encodings in the CRS model must inherently suffer from all of the drawbacks above.We give a new application to "big-key cryptography in the bounded-retrieval model", where secret keys are made intentionally huge to make them hard to exfiltrate. Using incompressible encodings, we can get all the security benefits of a big key without wasting storage space, by having the key to encode useful data.