Communication-Efficient Secure Computation of Encrypted Inputs using (k, n) Threshold Secret Sharing

Advancements in Internet of Things devices allow for the collection and analysis of big data. Moreover, cloud computing has emerged as an ideal platform for big data analysis because it quickly provides computing resources for analysis at scale. However, concerns exist regarding the privacy and security of this information. Secure computation is a technology that enables computation while maintaining data encryption. In this study, we present a new secure computation based on a client-server model, in which a set of servers perform computations using the inputs of multiple clients. We adopt the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(k, n)$ </tex-math></inline-formula> threshold secret sharing approach, where an input <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$s$ </tex-math></inline-formula> is divided into <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> shares and can be recovered from shares with a threshold <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> . However, conventional secure computation using <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(k,n)$ </tex-math></inline-formula> threshold secret sharing generally requires the condition <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n \geq 2k-1$ </tex-math></inline-formula> and communication among multiple servers for each multiplication. To the best of our knowledge, no previous study has solved this problem completely. We demonstrate that processes that require communication can be concentrated in the preprocessing phase and propose a secure computation using <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(k,n)$ </tex-math></inline-formula> threshold secret sharing that does not require communication during the secure computation phase, even when multiplication is performed. Moreover, we show that the number of communications does not depend on the number of multiplications, unlike in conventional methods. As communication often requires more processing time than the actual secure computation, our method makes it possible to realize faster overall processing than conventional methods. We also present an extensive security analysis and experimental simulation of the proposed method. Our proposed method achieves information-theoretic security against semi-honest adversaries under certain conditions with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n &lt; 2k-1$ </tex-math></inline-formula> .