Abstract
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 entire ciphertext. 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 n and security parameter k achieves 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 = ω(k 2 + ε), we can decode a bit of the plaintext from a corrupted ciphertext while doing computation sublinear in n.
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Hemenway, B., Ostrovsky, R. (2008). Public-Key Locally-Decodable Codes. In: Wagner, D. (eds) Advances in Cryptology – CRYPTO 2008. CRYPTO 2008. Lecture Notes in Computer Science, vol 5157. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85174-5_8
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