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Self-dual normal basis is efficient for the arithmetic operations in extension field and especially trace calculation. However, self-dual normal bases do not exist in F p m when characteristic p is odd and degree m is even. This paper proposes a method to construct an efficient normal basis for trace calculation when extension degree is even. In this paper, we call it pseudo self-dual normal basis.

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Ensuring the reliability of data gathering from every connected device is an essential issue for promoting the advancement of the next paradigm shift, i.e., Industry 4.0. Blockchain technology is becoming recognized as an advanced tool. However, data collaboration using blockchain has not progressed sufficiently among companies in the industrial supply chain (SC) that handle sensitive data, such as those related to product quality, etc. There are two reasons why data utilization is not sufficiently advanced in the industrial SC. The first is that manufacturing information is top secret. Blockchain mechanisms, such as Bitcoin, which uses PKI, require plaintext to be shared between companies to verify the identity of the company that sent the data. Another is that the merits of data collaboration between companies have not been materialized. To solve these problems, this paper proposes a business-to-business collaboration system using homomorphic encryption and blockchain techniques. Using the proposed system, each company can exchange encrypted confidential information and utilize the data for its own business. In a trial, an equipment manufacturer was able to identify the quality change caused by a decrease in equipment performance as a cryptographic value from blockchain and to identify the change one month earlier without knowing the quality value.

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