Constructions Of Public-key Encryption Schemes Research Articles

On the Hardness of Sparsely Learning Parity with Noise

Abstract The Learning Parity with Noise (LPN) problem represents the average-case analogue of the NP-Complete problem “decoding linear codes”, and it has been extensively studied in learning theory, coding theory and cryptography with applications to quantum-resistant cryptographic schemes. However, LPN also suffers from large public key size which is the common drawback that hinders code-based cryptography from being practical. In this paper, we study a sparse variant of LPN whose public matrix consists of sparse vectors instead of following uniform distribution. We show a win–win argument that at least one of the following assumption is true: (i) either the hardness of sparse LPN is implied by that of the standard LPN under the same noise rate; (ii) or there exists new black-box constructions of public-key encryption schemes and oblivious transfer protocols from standard LPN. Since the second assumption relies on the infeasible noise regimes for LPN-based public-key cryptography, we believe that the first assumption is more likely to hold, i.e. sparse LPN is as hard as standard LPN. Finally, we give a (heuristic) method to further compress the sparse public matrix by evaluating pseudorandom functions with keys made public, whose security again resorts to the aforementioned win–win technique.

Chosen-Ciphertext Security via Correlated Products

We initiate the study of one-wayness under correlated products. We are interested in identifying necessary and sufficient conditions for a function f and a distribution on inputs $(x_1,\dots,x_k)$ so that the function $(f(x_1),\dots,f(x_k))$ is one-way. The main motivation of this study is the construction of public-key encryption schemes that are secure against chosen-ciphertext attacks (CCAs). We show that any collection of injective trapdoor functions that is secure under a very natural correlated product can be used to construct a CCA-secure public-key encryption scheme. The construction is simple, black-box, and admits a direct proof of security. It can be viewed as a simplification of the seminal work of Dolev, Dwork, and Naor [SIAM J. Comput., 30 (2000), pp. 391–437], while relying on a seemingly incomparable assumption. We provide evidence that security under correlated products is achievable by demonstrating that lossy trapdoor functions [Peikert and Waters, Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008, pp. 187–196] yield injective trapdoor functions that are secure under the above-mentioned correlated product. Although we currently base security under correlated products on existing constructions of lossy trapdoor functions, we argue that the former notion is potentially weaker as a general assumption. Specifically, there is no fully black-box construction of lossy trapdoor functions from trapdoor functions that are secure under correlated products.