Degree-D Reverse Multiplication-Friendly Embeddings

doi:10.1109/TIT.2025.3596305

Degree-D Reverse Multiplication-Friendly Embeddings

2025-10-01

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DOI：10.1109/TIT.2025.3596305

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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 Z(pk )and fields like F-pk , 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.

Keywords:

Polynomials

Zinc

Vectors

Finite element analysis

Protocols

Interpolation

Galois fields

Decoding

Artificial intelligence

Training

Reverse multiplication-friendly embedding

polynomial interpolation

function field

algebraic geometry

packing method

Degree-D Reverse Multiplication-Friendly Embeddings

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