Paper 2024/1307

On Algebraic Homomorphic Encryption and its Applications to Doubly-Efficient PIR

Abstract

The Doubly-Efficient Private Information Retrieval (DEPIR) protocol of Lin, Mook, and Wichs (STOC'23) relies on a Homomorphic Encryption (HE) scheme that is algebraic, i.e., whose ciphertext space has a ring structure that matches the homomorphic operations. While early HE schemes had this property, modern schemes introduced techniques to manage noise growth. This made the resulting schemes much more efficient, but also destroyed the algebraic property. In this work, we study the properties of algebraic HE and try to make progress in solving this problem. We first prove a lower bound of $2^{\Omega(2^d)}$ for the ciphertext ring size of algebraic HE schemes (in terms of the depth $d$ of the evaluated circuit), which demonstrates a gap between optimal algebraic HE and the existing schemes, which have a ciphertext ring size of $2^{O(2^{2d})}$. As we are unable to bridge this gap directly, we instead slightly relax the notion of being algebraic. This allows us to construct a practically more efficient \emph{relaxed-algebraic} HE scheme. We then show that this also leads to a more efficient instantiation and implementation of DEPIR. We experimentally demonstrate run-time improvements of more than $4$x and reduce memory queries by more than $8$x compared to prior work. Notably, our relaxed-algebraic HE scheme relies on a new variant of the Ring Learning with Errors (RLWE) problem that we call $\{0, 1\}$-CRT RLWE. We give a formal security reduction to standard RLWE, and estimate its concrete security. Both the $\{0, 1\}$-CRT RLWE problem and the techniques used for the reduction may be of independent interest.