• Journal of KIISE:Computer Systems and Theory
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• v.30 no.5_6
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• pp.324-329
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• 2003

The public-key cryptosystems such as Diffie-Hellman Key Distribution and Elliptical Curve Cryptosystems are built on the basis of the operations defined in GF(2$^{m}$ ):addition, subtraction, multiplication and multiplicative inversion. It is important that these operations should be computed at high speed in order to implement these cryptosystems efficiently. Among those operations, as being the most time-consuming, multiplicative inversion has become the object of lots of investigation Formant's theorem says $\beta$$^{-1}$ =$\beta$$^{2}$sup m/-2/, where $\beta$$^{-1}$ is the multiplicative inverse of $\beta$$\in$GF(2$^{m}$ ). Therefore, to compute the multiplicative inverse of arbitrary elements of GF(2$^{m}$ ), it is most important to reduce the number of times of multiplication by decomposing 2$^{m}$ -2 efficiently. Among many algorithms relevant to the subject, the algorithm proposed by Itoh and Tsujii[2] has reduced the required number of times of multiplication to O(log m) by using normal basis. Furthermore, a few papers have presented algorithms improving the Itoh and Tsujii's. However they have some demerits such as complicated decomposition processes[3,5]. In this paper, in the case of 2$^{m}$ -2, which is mainly used in practical applications, an efficient algorithm is proposed for computing the multiplicative inverse at high speed by using both the factorization formula x$^3$-y$^3$=(x-y)(x$^2$＋xy＋y$^2$) and normal basis. The number of times of multiplication of the algorithm is smaller than that of the algorithm proposed by Itoh and Tsujii. Also the algorithm decomposes 2$^{m}$ -2 more simply than other proposed algorithms.

• Economic and Environmental Geology
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• v.8 no.1
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• pp.1-23
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• 1975

The subject of this study deals with phase relations between stannite ($Cu_2FeSnS_4$) and sphalerite (${\beta}-ZnS$)/wurtzite (${\alpha}-ZnS$). The phase relations were systematically investigated from liquidus temperature to $400^{\circ}C$ under controlled conditions. ${\beta}-stannite$ (tetragonal) is stable up to $706{\pm}5^{\circ}C$, where it inverts to a high-temperature polymorph ${\alpha}-stannite$ (cubic) melting congruently at $867{\pm}5^{\circ}C$. Sphalerite (cubic, ${\beta}-ZnS$) inverts at $1013{\pm}3^{\circ}C$ to wurtzite, which is the hexagonal hightemperature polymorph of ZnS. Between ${\alpha}-stannite$ and sphalerite a complete solid solution series exists above approximately $870^{\circ}C$ up to solidus temperature. The melting temperature of ${\alpha}-stannite$ rises towards sphalerite and reaches a maximum at $1074{\pm}3^{\circ}C$, which is the peritectic with the composition of 91 wt. % sphalerite and 9 wt. % ${\alpha}-stannite$. At this temperature, wurtzite takes only 5wt. % ${\alpha}-stannite$ in solid solution which decreases with increasing temperature. The inverson temperature of ${\alpha}/{\beta}-stannite$ is lowered with increasing amounts of sphalerite in solid solution down to $614{\pm}7^{\circ}C$, which is the eutectoid with the composition of 13 wt. % sphalerite and 87 wt. % ${\alpha}-stannite$. Here, ${\beta}-stannite$ contains only 10wt. % sphalerite in solid solution. With decreasing temperature, the ranges of the solid solution on both sides of the system narrow. The phase relations in the above pure system changed due to the FeS impurities in the sphalerite solid solution. The eutectoid increased from $614{\pm}7^{\circ}C$ up to $695{\pm}5^{\circ}C$ (5 wt. % FeS) and $700{\pm}5^{\circ}C$ (10wt. % FeS), while the peritectic decreased from $1074{\pm}3^{\circ}C$ down to $1036{\pm}3^{\circ}C$ (wt. %FeS) and $987{\pm}3^{\circ}C$ (10wt. %FeS). A most notable change is the appearance of non-binary regions. An important feature is the combination of this study system with the experimental results reported by Sprinfer (1972). If a stannite-kesterite solid solution is used in the place of stannite as a bulk composition, the inversion temperature is lowered to less than $400^{\circ}C$ which belongs to temperatures of the hydrothermal region.