Public-Key Encryption with Efficient Amortized Updates

Abstract

Searching and modifying public-key encrypted data has received a lot of attention in recent literature. In this paper we re-visit this important topic and achieve improved amortized bounds including resolving a prominent open question posed by Boneh et al. [3].

First, we consider the following much simpler to state problem: A server holds a copy of Alice’s database that has been encrypted under Alice’s public key. Alice would like to allow other users in the system to replace a bit of their choice in the server’s database by communicating directly with the server, despite other users not having Alice’s private key. However, Alice requires that the server should not know which bit was modified. Additionally, she requires that the modification protocol should have “small” communication complexity (sub-linear in the database size). This task is referred to as private database modification, and is a central tool in building a more general protocol for modifying and searching over public-key encrypted data. Boneh et al. [3] first considered the problem and gave a protocol to modify 1 bit of an N-bit database with communication complexity $\mathcal{O}(\sqrt N)$. Naturally, one can ask if we can improve upon this. Indeed, the recent work of Gentry [9] shows that under lattice assumptions, better asymptotic communication complexity is possible. However, current algebraic techniques based on any singly homomorphic encryption, or bilinear maps (which includes for example, all known cryptosystems based on factoring and discrete logs) cannot achieve communication better than $\mathcal{O}(\sqrt N)$ (see [17]). In this paper we study the problem of improving the communication complexity for modifying L bits of an N-bit database. Our main result is a black-box construction of a private database modification protocol to modify L bits of an N-bit database, using a protocol for modifying 1 bit. Our protocol has communication complexity $\tilde{\mathcal{O}}(N^\beta L^{(1+\alpha)(1-\beta)})$, where 0 < α< 1 can be an arbitrary constant and N ^β, 0 < β< 1 (for constant β) is the communication complexity of a protocol for modifying 1 bit of an N-bit database. We stress that our amortized protocol improves the communication complexity in all cases when the single bit modification protocol uses any known cryptosystem based on factoring or discrete logs.

In addition to our general reduction, we show how to realize an implementation of our amortized protocol under the subgroup decision problem [2]. (We remark that in contrast with recent work of Lipmaa [16] on the same topic, our database size does not grow with every update, and stays exactly the same size.)

As sample corollaries to our main result, we obtain the following:

• First, we apply our private database modification protocol to answer the main open question of [3]. More specifically, we construct a public-key encryption scheme supporting PIR queries that allows every message to have a non-constant number of keywords associated with it, which is secure under the subgroup decision problem.

• Second, we show that one can apply our techniques to obtain more efficient communication complexity when parties wish to increment or decrement multiple cryptographic counters (formalized by Katz et al. [15]).

We believe that “public-key encrypted” amortized database modification is an important cryptographic primitive in its own right and will be useful in other applications.