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Олег Тараскин @Crittografo

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Михаил, я написал вам в личные сообщения
Исправил — вроде теперь видно.
Если вам интересны конечные поля и вам нужны примеры, чтобы разобраться, то рекомендую установить open-source пакет SAGE — он много чего умеет. А кроме того, я бы почитал классику: «Конечные поля» Лиддла и Нидеррайтера или еще лучше «Алгебра» Глухова
Нет, не будет. По модулю 257 будет.
Да, конечно :)

Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 2 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 3 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 7 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 11 over Finite Field of size 19
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12
Elliptic Curve defined by y^2 = x^3 + 14 over Finite Field of size 19
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13
Elliptic Curve defined by y^2 = x^3 + x + 2 over Finite Field of size 19
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12
Elliptic Curve defined by y^2 = x^3 + x + 5 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + x + 6 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + x + 7 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + x + 10 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + x + 11 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 2*x + 6 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 2*x + 8 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 2*x + 10 over Finite Field of size 19
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17
Elliptic Curve defined by y^2 = x^3 + 2*x + 12 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 2*x + 14 over Finite Field of size 19
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14
Elliptic Curve defined by y^2 = x^3 + 2*x + 15 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 2*x + 17 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 2*x + 18 over Finite Field of size 19
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13
Elliptic Curve defined by y^2 = x^3 + 3*x + 6 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 3*x + 8 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 3*x + 10 over Finite Field of size 19
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17
Elliptic Curve defined by y^2 = x^3 + 3*x + 12 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 3*x + 14 over Finite Field of size 19
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14
Elliptic Curve defined by y^2 = x^3 + 3*x + 15 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 3*x + 17 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 3*x + 18 over Finite Field of size 19
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13
Elliptic Curve defined by y^2 = x^3 + 4*x + 1 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 4*x + 3 over Finite Field of size 19
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12
Elliptic Curve defined by y^2 = x^3 + 4*x + 7 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 4*x + 9 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 4*x + 15 over Finite Field of size 19
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14
Elliptic Curve defined by y^2 = x^3 + 4*x + 17 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 5*x + 1 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 5*x + 4 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 5*x + 11 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 5*x + 13 over Finite Field of size 19
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14
Elliptic Curve defined by y^2 = x^3 + 5*x + 14 over Finite Field of size 19
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12
Elliptic Curve defined by y^2 = x^3 + 5*x + 16 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 6*x + 1 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 6*x + 3 over Finite Field of size 19
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12
Elliptic Curve defined by y^2 = x^3 + 6*x + 7 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 6*x + 9 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 6*x + 15 over Finite Field of size 19
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14
Elliptic Curve defined by y^2 = x^3 + 6*x + 17 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 7*x + 2 over Finite Field of size 19
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12
Elliptic Curve defined by y^2 = x^3 + 7*x + 5 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 7*x + 6 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 7*x + 7 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 7*x + 10 over Finite Field of size 19
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14
Elliptic Curve defined by y^2 = x^3 + 7*x + 11 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 8*x + 2 over Finite Field of size 19
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14
Elliptic Curve defined by y^2 = x^3 + 8*x + 8 over Finite Field of size 19
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13
Elliptic Curve defined by y^2 = x^3 + 8*x + 9 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 8*x + 12 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 8*x + 13 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 8*x + 15 over Finite Field of size 19
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17
Elliptic Curve defined by y^2 = x^3 + 8*x + 16 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 8*x + 18 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 9*x + 1 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 9*x + 3 over Finite Field of size 19
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12
Elliptic Curve defined by y^2 = x^3 + 9*x + 7 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 9*x + 9 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 9*x + 15 over Finite Field of size 19
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14
Elliptic Curve defined by y^2 = x^3 + 9*x + 17 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 10*x + 3 over Finite Field of size 19
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14
Elliptic Curve defined by y^2 = x^3 + 10*x + 4 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 10*x + 5 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 10*x + 8 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 10*x + 10 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 10*x + 12 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 10*x + 13 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 10*x + 18 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 11*x + 2 over Finite Field of size 19
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12
Elliptic Curve defined by y^2 = x^3 + 11*x + 5 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 11*x + 6 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 11*x + 7 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 11*x + 10 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 11*x + 11 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 12*x + 2 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 12*x + 8 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 12*x + 9 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 12*x + 12 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 12*x + 13 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 12*x + 15 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 12*x + 16 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 12*x + 18 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 13*x + 3 over Finite Field of size 19
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14
Elliptic Curve defined by y^2 = x^3 + 13*x + 4 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 13*x + 5 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 13*x + 8 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 13*x + 10 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 13*x + 12 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 13*x + 13 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 13*x + 18 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 14*x + 6 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 14*x + 8 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 14*x + 10 over Finite Field of size 19
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17
Elliptic Curve defined by y^2 = x^3 + 14*x + 12 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 14*x + 14 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 14*x + 15 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 14*x + 17 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 14*x + 18 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 15*x + 3 over Finite Field of size 19
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14
Elliptic Curve defined by y^2 = x^3 + 15*x + 4 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 15*x + 5 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 15*x + 8 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 15*x + 10 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 15*x + 12 over Finite Field of size 19
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13
Elliptic Curve defined by y^2 = x^3 + 15*x + 13 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 15*x + 18 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 16*x + 1 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 16*x + 4 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 16*x + 11 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 16*x + 13 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 16*x + 14 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 16*x + 16 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 17*x + 1 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 17*x + 4 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 17*x + 11 over Finite Field of size 19
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18
Elliptic Curve defined by y^2 = x^3 + 17*x + 13 over Finite Field of size 19
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14
Elliptic Curve defined by y^2 = x^3 + 17*x + 14 over Finite Field of size 19
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12
Elliptic Curve defined by y^2 = x^3 + 17*x + 16 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 18*x + 2 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 18*x + 8 over Finite Field of size 19
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13
Elliptic Curve defined by y^2 = x^3 + 18*x + 9 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 18*x + 12 over Finite Field of size 19
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15
Elliptic Curve defined by y^2 = x^3 + 18*x + 13 over Finite Field of size 19
has order
16
Elliptic Curve defined by y^2 = x^3 + 18*x + 15 over Finite Field of size 19
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Elliptic Curve defined by y^2 = x^3 + 18*x + 16 over Finite Field of size 19
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16
Elliptic Curve defined by y^2 = x^3 + 18*x + 18 over Finite Field of size 19
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где p — это простое число. Но даже если p = 2 и число элементов получится 2^n, то все равно вам придется взять в одну из двух частей 0 или 1 и они уже не будут полями.
Да не вопрос, сегодня-завтра добавлю материал про группы и поле с примерами.
Допустим, что у вас есть конечное поле из p^n элементов.(все конечные поля имеют такой порядок)
Вы не можете взять и поделить его на две одинаковые части, поскольку каждая из них по отдельности не будет полем.
Объясните пожалуйста, что такое поле [1,2,3,4,5,6]? вы не забыли про 0? :)
«Все точки кривой» — это множество всех точек кривой. Понятие «группа точек кривой» предполагает наличие операции между любыми двумя точками, результат которой — точка той же самой кривой. Эта операция должна удовлетворять определенным правилам, которые одинаковы для всех групп (мультипликативная группа, которая используется в классическом Диффи-Хеллмане также этим правилам удовлетворяет (иначе она называлась бы не группа)). Вот эти правила: 1. ассоциативность (a*b)*c = a*(b*c) 2. в группе всегда есть нейтральный элемент e такой, что a*e = e*a = a (для любого a). 3. для любого элемента x в группе существует обратный элемент y: x*y = y*x = e
При написании статьи всегда возникает вопрос: насколько далеко стоит уходить в описание элементарных понятий. Ведь можно сделать статью либо недоступной к пониманию, либо просто скучной. Ее главная цель — вызвать интерес к предмету, даже если что-то не совсем понятно. В любом случае большое спасибо за сигнал — я добавлю определений в первую статью.
Пожалуйста! Рад, что вам понравилась статья. Видимо не зря старался :)
Порядок группы — это число ее элементов. Порядок группы точек кривой — число элементов группы точек кривой.
Но можно сказать еще проще: это число всех точек на конкретной кривой + одна точка на бесконечности. Вы можете посмотреть мою предыдущую статью

Конечно важно, но это скорее революция в других областях как минимум в финансах. С точки зрения криптографии в блокчейне по сравнению с изобретением асимметрии ничего принципиально нового нет.

Здравствуйте, Scratch! В самое ближайшее время опубликуем статью именно про изогении. В первую часть я решил про них не писать, так как в результате статья получилась бы слишком длинной. Ведь как сказал, кажется, Стивен наш Хокинг, каждая формула уменьшает число читателей вдвое.

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