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This paper proposes the computation of the Tate pairing, Ate pairing and its variations on the special Jacobi quartic elliptic curve Y2=dX4+Z4. We improve the doubling and addition steps in Miller's algorithm to compute the Tate pairing. We use the birational equivalence between Jacobi quartic curves and Weierstrass curves, together with a specific point representation to obtain the best result to date among curves with quartic twists. For the doubling and addition steps in Miller's algorithm for the computation of the Tate pairing, we obtain a theoretical gain up to 27% and 39%, depending on the embedding degree and the extension field arithmetic, with respect to Weierstrass curves and previous results on Jacobi quartic curves. Furthermore and for the first time, we compute and implement Ate, twisted Ate and optimal pairings on the Jacobi quartic curves. Our results are up to 27% more efficient compared to the case of Weierstrass curves with quartic twists.

uniformizer t1 at the point at infinity on the curve. Recall that another way to compute f .P /, when f is non-zero and defined at a point P , is as the leading coefficient of a series expansion of f with respect to a uniformizer of the curve at P . More precisely, if we can write f D a0Ca1t1Ca2t21C then we have f .1/ D a0. Following [5] we will insist that all our Miller functions fr;D1 are normalized at infinity; meaning that fr;D1 D t i 1 C aiC1t iC1 1 C for some i 2 Z. Another way to express this is that the leading coefficient of the power series expansion of fr

PSC-CUNY Research Award Program. 26 R. Miller functions according to the difficulty of computing them. Therefore, computable model theory investigates the feasibility of deciding facts or computing functions on mathematical structures. This article, aimed at group theorists having inter- est in issues of decidability, is written to introduce the basic definitions used in computable model theory, and to give a sample of the sort of results which fol- low from these concepts. Instead of applying computability to model theory in general, however, we will restrict

the Best Actor Oscar), Philadelphia is never really Beckett's story. Instead, his plight is merely a way to depict the transformation of Miller, the straight lawyer who represents Middle America's homophobia. The relationship between Beckett and Miller functions as a metaphor for there­ lationship between Middle America and gay white men. Throughout the film, Joe Miller is the mouthpiece for people's fear and hatred of homosexuals. Similar to films on racism in which the focus is on the prejudiced white person's transformation rather than on the African

imaginary, and the real are im- bricated. Operant in each of the three registers, the Miller functions as the core or knot around which the Reeve constructs himself. If the Miller’s body is more imposing, more there, than the Reeve’s, so, too, is his virility.³⁵ Perhaps because he has the story of Samson in mind, Chaucer-the- pilgrim uses hair to suggest strength, vigor, and sexual potency. In describing the Miller’s hirsute features, he has this to say: “His berd as any sowe or fox was reed, / And therto brood, as though it were a spade” (I (A), 552–53). Even the wart on

. 93 ZIMMERMANN, Texte 410 sieht in der Charakteristik Melchisedeks als Engelsgestalt (bzw. eine eindeutige Unterscheidung vom Gesalbten, Engel sind keine Gesalbten; der Gesalbte ist als Mensch im irdischen Bereich vorzustellen, Melchisedek im Himmel tätig. 94 Vgl. zu dieser Textrekonstruktion DE JONGE/VAN DER WOUDE, 1 lQMelchizedek 302; FLTZMYER, Further Light 252f.265f. (ohne Artikel vor ITDD). 95 Zu Jes 61,1 f. in Verbindung mit Lev 25 als Zentraltexte in llQMelch (die durch litur- gischen Gebrauch verbunden waren) vgl. BROOKE, Exegesis 321-323; MILLER

. Though the mills were owned by the monasteries, they were not operated by the monks but by lay families known as wei-hu (miller families) chosen from among the tenant farmers on the monastic estates. Besides the millers, there were also maintenance men referred to as wei-po-shih who were very likely technicians or skilled workers responsible for keeping the mills in good running order. Strictly speaking, the miller's function was to provide flour to feed the monks in the monastery. It must be kept in mind that we are dealing with people in north China, whose