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70:) for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results for
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As in the case of elliptic curves, explicit formulae for this pairing can be given in terms of
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1211:
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71:
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The Weil pairing does not extend to a pairing on all the torsion points (the direct limit of
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16:
Binary function non degenerative defined between the point of twist of an abelian variety
44:
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63:
55:
32:
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166:
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A down-to-earth construction of the Weil pairing is as follows. Choose a function
1277:
744:
20:
699:
is alternating and bilinear, giving rise to a non-degenerate pairing on the
580:{\displaystyle \mathrm {div} (F)=\sum _{0\leq k<n}-\sum _{0\leq k<n}.}
695:
times must give 1) other than 1. With this definition it can be shown that
1131:
768:(ÎĽ) of the multiplicative group (the inverse limit of â„“ roots of unity).
58:. More generally there is a similar Weil pairing between points of order
28:
1249:(1940), "Sur les fonctions algébriques à corps de constantes fini",
1048:, which in this particular case happens to be an isomorphism (see
714:
are not the same. However they do fit together to give a pairing
761:(the inverse limit of the â„“-torsion points) to the Tate module
1024:
is a projective, nonsingular curve of genus ≥ 0 over
844:{\displaystyle A\times A^{\vee }\longrightarrow \mu _{n}}
62:
of an abelian variety and its dual. It was introduced by
954:
then composition gives a (possibly degenerate) pairing
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is well-defined up to multiplication by a constant. If
74:
were known, and can be expressed simply by use of the
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with the polarisation gives a nondegenerate pairing
710:-torsion points) because the pairings for different
943:{\displaystyle \lambda :A\longrightarrow A^{\vee }}
1108:
1010:{\displaystyle A\times A\longrightarrow \mu _{n}.}
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424:
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153:
1109:{\displaystyle J\times J\longrightarrow \mu _{n}}
361:{\displaystyle E(K)=\{T\in E(K)\mid n\cdot T=O\}}
425:{\displaystyle \mu _{n}=\{x\in K\mid x^{n}=1\}}
784:, the Weil pairing is a nondegenerate pairing
1252:Les Comptes rendus de l'Académie des sciences
8:
419:
388:
355:
316:
1279:The Weil pairing on elliptic curves over C
1187:Homomorphic Signatures for Network Coding
1100:
1064:
1052:). Hence, composing the Weil pairing for
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1198:
606:if these points are all distinct. Then
677:{\displaystyle w(P,Q):={\frac {G}{F}}}
626:has the same divisor, so the function
7:
1044:induces a principal polarisation of
67:
1237:, available at www.jmilne.org/math/
780:over an algebraically closed field
772:Generalisation to abelian varieties
691:-th root of unity (as translating
602:, and a simple pole at each point
475:
472:
469:
221:{\displaystyle w(P,Q)\in \mu _{n}}
154:{\displaystyle E({\overline {K}})}
39:) on the points of order dividing
14:
1208:The Arithmetic of Elliptic Curves
594:has a simple zero at each point
1154:, and has also been applied in
1123:prime to the characteristic of
858:prime to the characteristic of
173:. The Weil pairing produces an
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113:) > 0) such that
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1210:. New York: Springer-Verlag.
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1156:elliptic curve cryptography
278:{\displaystyle P,Q\in E(K)}
119:primitive nth root of unity
1322:
1306:Pairing-based cryptography
1206:Silverman, Joseph (1986).
1177:Pairing-based cryptography
901:for higher dimensions. If
76:Weierstrass sigma function
1160:identity based encryption
882:{\displaystyle A^{\vee }}
86:Choose an elliptic curve
1146:The pairing is used in
1050:autoduality of Jacobians
897:. This is the so-called
757:) of the elliptic curve
633:Therefore if we define
622:, then by construction
37:multiplicative notation
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1011:
944:
883:
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678:
614:is the translation of
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105:to be coprime to char(
1182:Boneh–Franklin scheme
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235:, for any two points
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891:dual abelian variety
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905:is equipped with a
50:, taking values in
1152:algebraic geometry
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177:-th root of unity
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72:elliptic functions
1301:Abelian varieties
1235:Abelian Varieties
778:abelian varieties
687:we shall have an
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449:algebraic closure
163:Cartesian product
161:is known to be a
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97:, and an integer
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90:defined over a
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1272:External links
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45:elliptic curve
35:, though with
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1038:theta-divisor
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167:cyclic groups
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33:bilinear form
30:
26:
22:
1278:
1256:
1250:
1234:
1226:
1207:
1201:
1172:Tate pairing
1145:
1142:Applications
1135:
1129:
1124:
1120:
1118:
1053:
1045:
1041:
1029:
1025:
1021:
1019:
953:
907:polarisation
902:
899:Weil pairing
898:
894:
889:denotes the
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231:by means of
230:
174:
170:
125:-torsion on
122:
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102:
98:
94:
87:
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59:
51:
47:
40:
25:Weil pairing
24:
18:
1259:: 592–594,
1247:Weil, André
1231:James Milne
1036:, then the
745:Tate module
743:(ÎĽ) on the
121:. Then the
117:contains a
82:Formulation
21:mathematics
1290:Categories
1193:References
703:-torsion.
109:) if char(
64:André Weil
1098:μ
1094:⟶
1079:×
996:μ
992:⟶
977:×
936:∨
928:⟶
919:λ
875:∨
833:μ
829:⟶
815:∨
807:×
566:⋅
546:≤
539:∑
535:−
526:⋅
500:≤
493:∑
447:over the
401:∣
395:∈
377:μ
344:⋅
338:∣
323:∈
252:∈
210:μ
206:∈
169:of order
144:¯
1166:See also
1132:divisors
1119:for all
1034:Jacobian
854:for all
285:, where
1265:0002863
862:. Here
457:divisor
439:in the
165:of two
29:pairing
1263:
1214:
1028:, and
43:of an
27:is a
23:, the
1281:(PDF)
736:) →
455:with
92:field
1212:ISBN
1158:and
1150:and
1032:its
776:For
725:) Ă—
552:<
506:<
368:and
68:1940
1257:210
1134:of
1040:of
1020:If
893:of
628:G/F
618:by
590:So
451:of
443:of
54:th
19:In
1292::
1261:MR
1255:,
1233:,
1162:.
1138:.
1127:.
662::=
604:kQ
600:kQ
598:+
432:.
78:.
1220:.
1136:C
1125:k
1121:n
1102:n
1091:]
1088:n
1085:[
1082:J
1076:]
1073:n
1070:[
1067:J
1054:J
1046:J
1042:J
1030:J
1026:k
1022:C
1005:.
1000:n
989:]
986:n
983:[
980:A
974:]
971:n
968:[
965:A
950:,
932:A
925:A
922::
903:A
895:A
871:A
860:K
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837:n
826:]
823:n
820:[
811:A
804:]
801:n
798:[
795:A
782:K
766:â„“
763:T
759:E
755:E
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751:â„“
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741:â„“
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730:â„“
727:T
723:E
721:(
719:â„“
716:T
712:n
708:n
701:n
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693:n
689:n
670:F
667:G
659:)
656:Q
653:,
650:P
647:(
644:w
624:G
620:Q
616:F
612:G
608:F
596:P
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575:.
572:]
569:Q
563:k
560:[
555:n
549:k
543:0
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529:Q
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520:+
517:P
514:[
509:n
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489:=
486:)
483:F
480:(
476:v
473:i
470:d
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445:E
437:F
420:}
417:1
414:=
409:n
405:x
398:K
392:x
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386:=
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356:}
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