Properties of Primes and Multiplicative Group of a Field

In the [16] has been proven that the multiplicative group Z/pZ* is a cyclic group. Likewise, finite subgroup of the multiplicative group of a field is a cyclic group. However, finite subgroup of the multiplicative group of a field being a cyclic group has not yet been proven. Therefore, it is of importance to prove that finite subgroup of the multiplicative group of a field is a cyclic group. Meanwhile, in cryptographic system like RSA, in which security basis depends upon the difficulty of factorization of given numbers into prime factors, it is important to employ integers that are difficult to be factorized into prime factors. If both p and 2p + 1 are prime numbers, we call p as Sophie Germain prime, and 2p + 1 as safe prime. It is known that the product of two safe primes is a composite number that is difficult for some factoring algorithms to factorize into prime factors. In addition, safe primes are also important in cryptography system because of their use in discrete logarithm based techniques like Diffie-Hellman key exchange. If p is a safe prime, the multiplicative group of numbers modulo p has a subgroup of large prime order. However, no definitions have not been established yet with the safe prime and Sophie Germain prime. So it is important to give definitions of the Sophie Germain prime and safe prime. In this article, we prove finite subgroup of the multiplicative group of a field is a cyclic group, and, further, define the safe prime and Sophie Germain prime, and prove several facts about them. In addition, we define Mersenne number (Mn), and some facts about Mersenne numbers and prime numbers are proven.Arai Kenichi - Shinshu University, Nagano, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanBroderick Arneson and Piotr Rudnicki. Primitive roots of unity and cyclotomic polynomials. Formalized Mathematics, 12(1):59-67, 2004.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Yoshinori Fujisawa and Yasushi Fuwa. The Euler's function. Formalized Mathematics, 6(4):549-551, 1997.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relative primes. Formalized Mathematics, 1(5):829-832, 1990.Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97-104, 1991.Hiroyuki Okazaki and Yasunari Shidama. Uniqueness of factoring an integer and multiplicative group R/pZ*. Formalized Mathematics, 16(2):103-107, 2008, doi:10.2478/v10037-008-0015-1.Christoph Schwarzweller. The ring of integers, euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.Dariusz Surowik. Cyclic groups and some of their properties - part I. Formalized Mathematics, 2(5):623-627, 1991.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855-864, 1990.Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990

Formalization of the Advanced Encryption Standard. Part I

In this article, we formalize the Advanced Encryption Standard (AES). AES, which is the most widely used symmetric cryptosystem in the world, is a block cipher that was selected by the National Institute of Standards and Technology (NIST) as an official Federal Information Processing Standard for the United States in 2001 [12]. AES is the successor to DES [13], which was formerly the most widely used symmetric cryptosystem in the world. We formalize the AES algorithm according to [12]. We then verify the correctness of the formalized algorithm that the ciphertext encoded by the AES algorithm can be decoded uniquely by the same key. Please note the following points about this formalization: the AES round process is composed of the SubBytes, ShiftRows, MixColumns, and AddRoundKey transformations (see [12]). In this formalization, the SubBytes and MixColumns transformations are given as permutations, because it is necessary to treat the finite field GF(28) for those transformations. The formalization of AES that considers the finite field GF(28) is formalized by the future article.Arai Kenichi - Tokyo University of Science Chiba, JapanOkazaki Hiroyuki - Shinshu University Nagano, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.U.S. Department of Commerce/National Institute of Standards and Technology. FIPS PUB 197, Advanced Encryption Standard (AES). Federal Information Processing Standars Publication, 2001.Hiroyuki Okazaki and Yasunari Shidama. Formalization of the data encryption standard. Formalized Mathematics, 20(2):125-146, 2012. doi:10.2478/v10037-012-0016-y.Andrzej Trybulec. On the decomposition of finite sequences. Formalized Mathematics, 5 (3):317-322, 1996.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575-579, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Many argument relations. Formalized Mathematics, 1(4):733-737, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990

Polymerization of diacetylene phospolipid bilayers on solid substrate: Influence of the film deposition temperature

Micropatterned phospholipid bilayers on solid substrates offer an attractive platform for various applications, such as high throughput drug screening. We have previously developed a photopolymerization-based methodology for generating micropatterned bilayers composed of polymerized and fluid lipid bilayers. Lithographic photopolymerization of a diacetylene-containing phospholipid (DiynePC) allowed facile fabrication of compartmentalized arrays of fluid lipid membranes. Herein, we report on a key experimental parameter that significantly influences the homogeneity and quality of the fabricated polymeric bilayers, namely the temperature at which monolayers of monomeric DiynePC were formed on the water surface and transferred onto solid substrates by the Langmuir-Blodgett/Langmuir-Schaefer (LB/LS) technique. Using fluorescence microscopy and atomic force microscopy, it was found that polymerized bilayers were homogeneous, if bilayers of DiynePC were prepared below the triple point temperature (ca. 20 C) of the monolayer, where a direct transition from the gaseous state to the liquid condensed state occurred. Bilayers prepared above this temperature had a markedly increased number of crack-like line defects. The differences were attributed to the domain structures in the monolayer that were transferred from the water surface to the substrate. Domain size, rather than the molecular packing in each domain, was concluded to play a critical role in the formation of defects. The spontaneous curvature and area changes of bilayers were postulated to cause destabilization and detachment of the films from the substrate upon polymerization. Our present results highlight the importance of controlling the domain structures for the homogeneity of polymerized bilayers required in technological applications

The development of “Ultimate Rudder” for EEDI

EEDI (Energy Efficiency Design Index) came into effect mandatory in Jan. 2013, and the ship owners definitely required a higher efficiency propulsion system than ever before. Hence, the shipyards have been conducting an optimization of ESD (Energy Saving Device) system in self-propulsion test for each project. As the results, the shipyards have installed a rudder bulb as an effective ESD. The rudder bulb is a popular ESD system from a long time ago. Mewis1) described that the rudder bulb was developed by Costa in 1952 and the efficiency improve by the rudder bulb for a container vessel was 1% on average. Fujii et al.2) developed “MIPB (Mitsui Integrated Propeller Boss)” as an advanced rudder bulb. The feature of MIPB was a streamlined profile from propeller cap to rudder. According to their paper, the efficiency improve by installing MIPB was 2-4%. Recently, NAKASHIMA PROPELLER Co., Ltd. developed ECO-Cap (economical propeller cap)3) as a new ESD with FRP (Fiber Reinforced Plastics). The strength of FRP is higher than that of NAB (Nickel Aluminium Bronze), therefore ECO-Cap was able to adopt thin fins on propeller caps for low resistance. Although the material used for the energy- saving propeller cap was generally NAB, the research results on FRP showed that FRP could be used as ESD due to their properties such as lightweight and flexibility. As explained above, the authors thought that there was a possibility to evolve the rudder bulb profile using the easily moldable FRP compared with NAB. This paper described about the development of “Ultimate Rudder” of new design concept by FRP. The authors optimized the profile of “Ultimate Rudder” by CFD and confirmed the efficiency increase from 4.9 to 5.4% in self-propulsion test

Histo-chemical and cytochemical studies on the succinic dehydroge-nase system with three ditetrazolium salts, NT, Nitro-NT, and Nitro-Bt

1. Histochemical and cytochemical studies with respect to the sites of reaction were made on the succinic dehydrogenase system activity of human and animal tissues using ditetrazolium salts, namely, neotetrazolium chloride, nitro-neotetrazolium chloride, and nitra-blue tetrazolium chloride. 2. The advantages and disadvantages of each ditetrazolium salt for histochemical and cytochemical purposes and the reaction taking place in frozen tissue sections and that in fresh tissue blocks were compared, and the method of procedure suitable for each condition was established with some modification. 3. Selecting conditions suitable for cytochemical purpose, it was shown that the reaction took place at the sites coinciding with mitochondria, and the distribution of the enzyme reaction was also examined. In addition, several new findings in the brains and other tissues cytochemically made clear were pointed out.</p

N-Dimensional Binary Vector Spaces

This research was presented during the 2013 International Conference on Foundations of Computer Science FCS’13 in Las Vegas, USAThe binary set {0, 1} together with modulo-2 addition and multiplication is called a binary field, which is denoted by F2. The binary field F2 is defined in [1]. A vector space over F2 is called a binary vector space. The set of all binary vectors of length n forms an n-dimensional vector space Vn over F2. Binary fields and n-dimensional binary vector spaces play an important role in practical computer science, for example, coding theory [15] and cryptology. In cryptology, binary fields and n-dimensional binary vector spaces are very important in proving the security of cryptographic systems [13]. In this article we define the n-dimensional binary vector space Vn. Moreover, we formalize some facts about the n-dimensional binary vector space Vn.Arai Kenichi - Tokyo University of Science Chiba, JapanOkazaki Hiroyuki - Shinshu University Nagano, JapanJesse Alama. The vector space of subsets of a set based on symmetric difference. Formalized Mathematics, 16(1):1-5, 2008. doi:10.2478/v10037-008-0001-7.Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.X. Lai. Higher order derivatives and differential cryptoanalysis. Communications and Cryptography, pages 227-233, 1994.Robert Milewski. Associated matrix of linear map. Formalized Mathematics, 5(3):339-345, 1996.J.C. Moreira and P.G. Farrell. Essentials of Error-Control Coding. John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, 2006.Hiroyuki Okazaki and Yasunari Shidama. Formalization of the data encryption standard. Formalized Mathematics, 20(2):125-146, 2012. doi:10.2478/v10037-012-0016-y.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Wojciech A. Trybulec. Subspaces and cosets of subspaces in vector space. Formalized Mathematics, 1(5):865-870, 1990.Wojciech A. Trybulec. Linear combinations in vector space. Formalized Mathematics, 1 (5):877-882, 1990.Wojciech A. Trybulec. Basis of vector space. Formalized Mathematics, 1(5):883-885, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Many argument relations. Formalized Mathematics, 1(4):733-737, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Mariusz Zynel. The Steinitz theorem and the dimension of a vector space. Formalized Mathematics, 5(3):423-428, 1996

Normal Subgroup of Product of Groups

In [6] it was formalized that the direct product of a family of groups gives a new group. In this article, we formalize that for all j ∈ I, the group G = Πi∈IGi has a normal subgroup isomorphic to Gj. Moreover, we show some relations between a family of groups and its direct product.Okazaki Hiroyuki - Shinshu University, Nagano, JapanArai Kenichi - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127-134, 1998.Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855-864, 1990.Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990

Difference of Function on Vector Space over F

In [11], the definitions of forward difference, backward difference, and central difference as difference operations for functions on R were formalized. However, the definitions of forward difference, backward difference, and central difference for functions on vector spaces over F have not been formalized. In cryptology, these definitions are very important in evaluating the security of cryptographic systems [3], [10]. Differential cryptanalysis [4] that undertakes a general purpose attack against block ciphers [13] can be formalized using these definitions. In this article, we formalize the definitions of forward difference, backward difference, and central difference for functions on vector spaces over F. Moreover, we formalize some facts about these definitions.Arai Kenichi - Tokyo University of Science Chiba, JapanWakabayashi Ken - Shinshu University Nagano, JapanOkazaki Hiroyuki - Shinshu University Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.E. Biham and A. Shamir. Differential cryptanalysis of DES-like cryptosystems. Lecture Notes in Computer Science, 537:2-21, 1991.E. Biham and A. Shamir. Differential cryptanalysis of the full 16-round DES. Lecture Notes in Computer Science, 740:487-496, 1993.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.X. Lai. Higher order derivatives and differential cryptoanalysis. Communications and Cryptography, pages 227-233, 1994.Bo Li, Yan Zhang, and Xiquan Liang. Difference and difference quotient. Formalized Mathematics, 14(3):115-119, 2006. doi:10.2478/v10037-006-0014-z.Michał Muzalewski and Wojciech Skaba. From loops to Abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833-840, 1990.Hiroyuki Okazaki and Yasunari Shidama. Formalization of the data encryption standard. Formalized Mathematics, 20(2):125-146, 2012. doi:10.2478/v10037-012-0016-y.Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559-564, 2001.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992

Isomorphisms of Direct Products of Finite Cyclic Groups

In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.Arai Kenichi - Tokyo University of Science, Chiba, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. K¨onig’s theorem. Formalized Mathematics, 1(3):589-593, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841-845, 1990.Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181-187, 2005.Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Anna Lango and Grzegorz Bancerek. Product of families of groups and vector spaces. Formalized Mathematics, 3(2):235-240, 1992.Hiroyuki Okazaki, Noboru Endou, and Yasunari Shidama. Cartesian products of family of real linear spaces. Formalized Mathematics, 19(1):51-59, 2011, doi: 10.2478/v10037-011-0009-2.Christoph Schwarzweller. The ring of integers, Euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.Christoph Schwarzweller. Modular integer arithmetic. Formalized Mathematics, 16(3):247-252, 2008, doi:10.2478/v10037-008-0029-8.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990

Analysis of IGZO crystalline structure and its stability by first-principles calculations

In-Ga-Zn oxide (IGZO), an oxide semiconductor, has been actively researched as a semiconductor material having features different from those of silicon in recent years [1]. IGZO is used as a transistor material in backplanes of commercially available displays. Transistors including crystalline IGZO have high stability and thus are suitable for mass production [2]. Our previous studies revealed that the selected area diffraction pattern of an IGZO film formed at room temperature by sputtering is a halo pattern, whereas diffraction spots are observed in the diffraction pattern obtained by nanobeam electron diffraction with a probe diameter of 1 nm [3,4]. These results suggest that the IGZO film has rather nanometer-sized crystalline structures than a completely amorphous structure. We named this film “nano-crystalline IGZO (nc-IGZO) film.” Other researchers have reported that the nc-IGZO film has a crystalline-cluster composite structure, according to the analysis results obtained by grazing-incidence X-ray diffraction, anomalous X-ray scattering, and reverse-Monte-Carlo simulation [5]. In this study, an IGZO structure having a minute crystalline region, which was considered to exist in nc-IGZO as a local structure, was created by first-principles calculations and its stability was analyzed. The IGZO model having a crystalline region used in this study was obtained by a melt-quench method in the following manner. Note that the initial structure had a hexagonal-prism crystalline region at the center and an amorphous region (random atomic arrangement) around the crystalline region. The composition ratio was In:Ga:Zn:O = 1:1:1:4 and the density was 6.1 g/cm3. First, for structural relaxation with the crystalline region maintained, the amorphous region was fused in quantum molecular dynamics simulation (3500 K, 6 ps) while the atomic arrangement of the crystalline region was fixed, and the structure was cooled to 500 K at a rate of 500 K/ps and held at 300 K for 5 ps. Finally, the entire structure including the crystalline region was optimized towards the target structure (Fig. 1). An amorphous model was also created for reference. The amorphous model was obtained by quantum molecular dynamics simulation of the entire structure under similar temperature conditions without fixing the atomic arrangement of the crystalline region, followed by structural optimization. The comparison between the two models showed that the total energy of the IGZO model having a crystalline region was lower than that of the amorphous model (not having a crystalline region). This suggests that the crystalline region contributes to structure stabilization. Please click Additional Files below to see the full abstract