Abstract
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, so that element-wise multiplication in the former is somewhat “compatible” with the product in the latter. Then, several optimal bounds and results are presented, and furthermore, the concept is generalized from one multiplication to degrees larger than two. These packing methods encompass several constructions seen in the literature in contexts like secure multiparty computation and fully homomorphic encryption.
One such construction is the concept of reverse multiplication-friendly embeddings (RMFEs), which are essentially degree-2 packing methods. In this work we generalize the notion of RMFEs to degree-D RMFEs which, in spite of being “more algebraic” than packing methods, turn out to be essentially equivalent. Then, we present a general construction of degree-D RMFEs by generalizing the ideas on algebraic geometry used to construct traditional degree-2 RMFEs which, by the aforementioned equivalence, leads to explicit constructions of packing methods. 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.
Finally, we discuss interesting applications of our RMFEs, focusing in particular on the case of non-interactively generating high degree correlations for secure multiparty computation protocols. This requires the use of Shamir secret sharing for a large number of parties, which requires large-degree Galois ring extensions. Our RMFE enables the generation of such preprocessing data over small rings, without paying for the multiplicative overhead incurred by using Galois ring extensions of large degree. For our application we also construct along the way, as a side contribution of potential independent interest, a pseudo-random secret-sharing solution for non-interactive generation of packed Shamir-sharings over Galois rings with structured secrets, inspired by the PRSS solutions from (Benhamouda et al., TCC 2021).
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Notes
- 1.
In our actual definition, as in the definition of traditional (degree-2) RMFEs, the domain of \(\phi \)/codomain of \(\psi \) can be a Galois ring as well instead of \(\mathbb {Z}_{p^k}\).
- 2.
It is possible to improve this basic construction via certain concatenation techniques. However, any construction based on this polynomial evaluation cannot achieve constant ratio, which can be seen as an analogue of the concatenation of Reed-Solomon codes in the classic coding theory.
- 3.
\(c_P\) is only used for expressing divisor G explicitly so as to present the basic property of the function field. The explicit construction of G is not the focus of this paper. Thus, \(c_p\) will not appear in our construction.
- 4.
Such covering design sizes can be found in https://www.dmgordon.org/cover/.
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Acknowledgments
The work of Hongqing Liu was supported in part by the National Key Research and Development Program under the grant 2022YFA1004900. The work of Chaoping Xing was supported in part by the National Key Research and Development Project under the Grant 2021YFE0109900 and by the National Natural Science Foundation of China (NSFC) under the Grant 12031011. The work of Chen Yuan was supported in part by the National Natural Science Foundation of China under Grant 12101403.
This paper was prepared in part for information purposes by the Artificial Intelligence Research group of JPMorgan Chase & Co and its affiliates (“JP Morgan”), and is not a product of the Research Department of JP Morgan. JP Morgan makes no representation and warranty whatsoever and disclaims all liability, for the completeness, accuracy or reliability of the information contained herein. This document is not intended as investment research or investment advice, or a recommendation, offer or solicitation for the purchase or sale of any security, financial instrument, financial product or service, or to be used in any way for evaluating the merits of participating in any transaction, and shall not constitute a solicitation under any jurisdiction or to any person, if such solicitation under such jurisdiction or to such person would be unlawful. 2022 JP Morgan Chase & Co. All rights reserved.
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Escudero, D., Hong, C., Liu, H., Xing, C., Yuan, C. (2023). Degree-D Reverse Multiplication-Friendly Embeddings: Constructions and Applications. In: Guo, J., Steinfeld, R. (eds) Advances in Cryptology – ASIACRYPT 2023. ASIACRYPT 2023. Lecture Notes in Computer Science, vol 14438. Springer, Singapore. https://doi.org/10.1007/978-981-99-8721-4_4
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