Degree-D Reverse Multiplication-Friendly Embeddings

Abstract

Reverse multiplication-friendly embeddings have played a crucial role in secure multiparty computation and zero-knowledge proofs. In this work, we generalize the notion of RMFEs to degree-D RMFEs. We present a general construction of degree-D RMFEs by generalizing the ideas on algebraic geometry used to construct traditional degree-2 RMFEs. Furthermore, our theory is given in a unified manner for general Galois rings, which include both rings of the form $\mathbb {Z}_{p^{k}}$ and fields like $\mathbb {F}_{p^{k}}$ , which have been treated separately in prior works. We present multiple concrete sets of parameters for degree-D RMFEs (including $D=2$ ), which can be useful for future works. In the recent work of (Cheon & Lee, Eurocrypt’22), the concept of a degree-D packing method was formally introduced, which captures the idea of embedding multiple elements of a smaller ring into a larger ring. We show that the generalized notion of RMFEs to degree-D RMFEs which, in spite of being “more algebraic” than packing methods, turn out to be essentially equivalent. Thus, our constructions of degree-D RMFEs are also degree-D packing methods.