Motivated by applications in large storage systems, we initiate the study of incremental deterministic public-key encryption. Deterministic public-key encryption, introduced by Bellare, Boldyreva, and O'Neill (CRYPTO '07), provides an alternative to randomized public-key encryption in various scenarios where the latter exhibits inherent drawbacks. A deterministic encryption algorithm, however, cannot satisfy any meaningful notion of security for low-entropy plaintexts distributions, but Bellare et al. demonstrated that a strong notion of security can in fact be realized for relatively high-entropy plaintext distributions. In order to achieve a meaningful level of security, a deterministic encryption algorithm should be typically used for encrypting rather long plaintexts for ensuring a sufficient amount of entropy. This requirement may be at odds with efficiency constraints, such as communication complexity and computation complexity in the presence of small updates. Thus, a highly desirable property of deterministic encryption algorithms is incrementality: Small changes in the plaintext translate into small changes in the corresponding ciphertext. We present a framework for modeling the incrementality of deterministic public-key encryption. Our framework extends the study of the incrementality of cryptography primitives initiated by Bellare, Goldreich and Goldwasser (CRYPTO '94). Within our framework, we propose two schemes, which we prove to enjoy an optimal tradeoff between their security and incrementality up to lower-order factors. Our first scheme is a generic method which can be based on any deterministic public-key encryption scheme, and, in particular, can be instantiated with any semantically secure (randomized) public-key encryption scheme in the random-oracle model. Our second scheme is based on the Decisional Diffie-Hellman assumption in the standard model. The approach underpinning our schemes is inspired by the fundamental "sample-then-extract" technique due to Nisan and Zuckerman (JCSS '96) and refined by Vadhan (J. Cryptology '04), and by the closely related notion of "locally computable extractors" due to Vadhan. Most notably, whereas Vadhan used such extractors to construct private-key encryption schemes in the bounded-storage model, we show that techniques along these lines can also be used to construct incremental public-key encryption schemes.
