Abstract Balogh et al. proposed deterministic secure multiparty computation called private PEZ protocols . In their work, a general construction of private PEZ protocols for computing an arbitrary function with n inputs is presented, but the function’s inputs must be binary . Binary domains are sufficient for computing functions with arbitrary domains because we can use the binary expansion of the inputs. However, such an expansion makes the protocol inefficient because unnecessary privacy is considered among the expanded bits of each input. Hence, we remove the binary expansion technique in this paper and propose a new private PEZ protocol directly applicable to functions with arbitrary domains. The proposed private PEZ protocol for an arbitrary function is much more efficient than Balogh et al.’s protocol. Concretely, an efficiency measure called the length of an initial string is exponentially improved in the domain size m : the order obtained from our construction is $$\mathcal {O}((2m)^{m^{n-1}})$$ O ( ( 2 m ) m n - 1 ) , while the construction of Balogh et al. yields $$\mathrm {\Omega }\bigg (\bigg (2^{\frac{m}{2}}\bigg )^{m^{n-1}}\bigg )$$ Ω ( ( 2 m 2 ) m n - 1 ) . The key idea of our protocol is called the divide and cue strategy, based on a recursive structure of views in private PEZ protocols.

Several relative entropy-based methods have been studied in cyber-attack detection of control systems. Most existing studies set the threshold values of relative entropy by trial and error such that their error probabilities become small. Meanwhile, the relationship between threshold values and error probabilities in likelihood ratio tests is clarified by Information theory. Information theory also clarifies the relationship between relative entropy and likelihood ratio test. To theoretically set the threshold, the authors have investigated the relationship between relative entropy and the likelihood ratio test using experimental data from DoS attacks and man-in-the-middle attacks on control communication (Modbus TCP). This paper investigates the relationship between threshold values and error probabilities in actual experiments. Error probabilities are classified as false positive rates and false negative rates. Neyman-Pearson lemma shows how to construct a detector that considers the trade-off between false positive and false negative rates. Stein’s lemma shows how to give optimal threshold values. We build a detector from the two lemmas that consider the trade-off with probability models of delay time between Response and ACK of Modbus TCP. We conduct experiments and discuss optimal threshold-setting methods in the sense that the false positive rates cannot be further reduced when false positive rates are fixed.