Public-Key Locally-Decodable Codes

In this paper we introduce the notion of a Public-Key Encryption Scheme that is also a Locally-Decodable Error-Correcting Code (PKLDC). In particular, we allow any polynomial-time adversary to read the entire ciphertext, and corrupt a constant fraction of the bits of the entireciphertext. Nevertheless, the decoding algorithm can recover any bit of the plaintext with all but negligible probability by reading only a sublinear number of bits of the (corrupted) ciphertext. We give a general construction of a PKLDC from any Semantically-Secure Public Key Encryption (SS-PKE) and any Private Information Retrieval (PIR) protocol. Since Homomorphic encryption implies PIR, we also show a reduction from any Homomorphic encryption protocol to PKLDC. Applying our construction to the best known PIR protocol (that of Gentry and Ramzan), we obtain a PKLDC, which for messages of size nand security parameter kachieves ciphertexts of size $\mathcal{O}(n)$, public key of size $\mathcal{O}(n+k)$, and locality of size $\mathcal{O}(k^2)$. This means that for messages of length n= i¾?(k2 + i¾?), we can decode a bit of the plaintext from a corrupted ciphertext while doing computation sublinear in n.

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