• Communications of Mathematical Education
• /
• v.31 no.3
• /
• pp.279-290
• /
• 2017

This study proposes a self learning material for understanding the key contents of Galois theory. This material is for teachers who have learned algebraic structures like group, field, and vector space which are related with Galois theory but do not clearly understand how algebraic structures are related with the solvability of polynomials and school mathematics. This material is likely to help them to overcome such difficulties. Even though proposed material is used mainly for self learning, the teachers may be helped once or twice by some professionals. In this article, two expressions 'solvability of polynomial' and 'solvability of equation' have the same meaning and 'teacher' means in-service mathematics teacher.

• Journal of Power System Engineering
• /
• v.8 no.1
• /
• pp.62-68
• /
• 2004

This paper describes a fault tolerant control in distributed PLC(Programmable Logic Controller) system to ensure reliability of controllers which have some faults simultaneously. First, the behavior of PLC is modeled as discrete expressions using Galois field. Then, we design the control laws for additional spare controllers to generate parity code with two dimensions. Finally, the algorithm for estimating normal output instead of abnormal output from the controllers with fault is suggested. Comparing to the traditional duplication method, the suggested method can reduce the number of spare controllers significantly to ensure control reliability. This method will be applied to an automatic system in order to increase reliability. Also, it can improve cost performance of the system.

• Convergence Security Journal
• /
• v.18 no.4
• /
• pp.27-33
• /
• 2018

Stream cipher is an one-time-pad type encryption algorithm that encrypt plaintext using simple operation such as XOR with random stream of bits (or characters) as symmetric key and its security depends on the randomness of used stream. Therefore we can design more secure stream cipher algorithm by using mathematical analysis of the stream such as period, linear complexity, non-linearity, correlation-immunity, etc. The key stream in stream cipher is generated in linear feedback shift register(LFSR) having characteristic polynomial. The primitive polynomial is the characteristic polynomial which has the best security property. It is used widely not only in stream cipher but also in SEED, a block cipher using 8-degree primitive polynomial, and in Chor-Rivest(CR) cipher, a public-key cryptosystem using 24-degree primitive polynomial. In this paper we present the concept and various properties of primitive polynomials in Galois field and prove the theorem finding the number of irreducible polynomials and primitive polynomials over $F_p$ when p is larger than 2. This kind of research can be the foundation of finding primitive polynomials of higher security and developing new cipher algorithms using them.