Abstract
Private Simultaneous Messages (PSM) is a minimal model of secure computation, where the input players with shared randomness send messages to the output player simultaneously and only once. In this field, finding upper and lower bounds on communication complexity of PSM protocols is important, and in particular, identifying the optimal one where the upper and lower bounds coincide is the ultimate goal. However, up until now, functions for which the optimal communication complexity has been determined are few: An example of such a function is the two-input AND function where \((2\log _2 3)\)-bit communication is optimal. In this paper, we provide new upper and lower bounds for several concrete functions. For lower bounds, we introduce a novel approach using combinatorial objects called abstract simplicial complexes to represent PSM protocols. Our method is suitable for obtaining non-asymptotic explicit lower bounds for concrete functions. By deriving lower bounds and constructing concrete protocols, we show that the optimal communication complexity for the equality and majority functions with three input bits are \(3\log _2 3\) bits and 6 bits, respectively. We also derive new lower bounds for the n-input AND function, three-valued comparison function, and multiplication over finite rings.
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Acknowledgments
This work was supported by JSPS KAKENHI Grant Numbers JP19H01109, JP21K17702, JP22K11906, and JP23H00479, and JST CREST Grant Number JPMJCR22M1, Japan. This work was supported by Institute of Mathematics for Industry, Joint Usage/Research Center in Kyushu University. (FY2022 Short-term Visiting Researcher “On Minimal Construction of Private Simultaneous Messages Protocols” (2022a006) and FY2023 Short-term Visiting Researcher “On the Relationship between Physical and Non-physical Secure Computation Protocols” (2023a009)).
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Shinagawa, K., Nuida, K. (2024). Explicit Lower Bounds for Communication Complexity of PSM for Concrete Functions. In: Chattopadhyay, A., Bhasin, S., Picek, S., Rebeiro, C. (eds) Progress in Cryptology – INDOCRYPT 2023. INDOCRYPT 2023. Lecture Notes in Computer Science, vol 14460. Springer, Cham. https://doi.org/10.1007/978-3-031-56235-8_3
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