Revision history [back]
 | 1 | initial version |
An answer could be as follows: 'p=18446744073709551667' then define the field 'K.=GF(p^2)' over which we define the curve 'E = EllipticCurve(K,[1,0])'. Now use the code 'E.cardinality().factor()' to get the cardinality of $E$, which is $2^4 * 4999^2 * 922521708027083^2$. Use the code 'pts = E.gens()' to find the generators which turn out to be $((3967812587828368975a + 7437411822947458617 : 16672546567636985011a + 14299206839437184441 : 1)$, $ (8681109523785822829a + 7072963597633280041 : 15515941078240688329a + 11174851755365315891 : 1))$.
Let $P_1$ be the first point, that is 'P1=E(13308581970510900443a + 15361864644717267429, 12035351063104026383a + 6630581890190451124)' and use the code 'P1.order().factor()' to get the order of $P_1$, which is $2^2 * 4999 * 922521708027083$. Now, let 'Q1=2^2922521708027083P1', then $Q_1$ is of the required order. We can repeat this with the other point.
| | 2 | No.2 Revision |
An answer could be as follows: 'p=18446744073709551667' then define the field 'K.=GF(p^2)' over which we define the curve 'E = EllipticCurve(K,[1,0])'. Now use the code 'E.cardinality().factor()' to get the cardinality of $E$, which is $2^4 * 4999^2 * 922521708027083^2$. Use the code 'pts = E.gens()' to find the generators which turn out to be $((3967812587828368975a + 7437411822947458617 : 16672546567636985011a + 14299206839437184441 : 1)$, $ (8681109523785822829a + 7072963597633280041 : 15515941078240688329a + 11174851755365315891 : 1))$.
Let $P_1$ be the first point, that is
'P1=E(13308581970510900443a + 15361864644717267429, 12035351063104026383a + 6630581890190451124)'
and use the code
'P1.order().factor()'
to get the order of $P_1$, which is $2^2 * 4999 * 922521708027083$. Now, let 'Q1=2^2922521708027083'Q1=2^2 * 922521708027083 * P1', then $Q_1$ is of the required order. We can repeat this with the other point. point.
| | 3 | No.3 Revision |
An answer could be as follows:
'p=18446744073709551667'
then define the field
'K.=GF(p^2)'
'K . < a >=GF(p^2)'
over which we define the curve
'E = EllipticCurve(K,[1,0])'. Now use the code
'E.cardinality().factor()' to get the cardinality of $E$, which is $2^4 * 4999^2 * 922521708027083^2$.
Use the code 'pts = E.gens()' to find the generators which turn out to be $((3967812587828368975a + 7437411822947458617 : 16672546567636985011a + 14299206839437184441 : 1)$, $ (8681109523785822829a + 7072963597633280041 : 15515941078240688329a + 11174851755365315891 : 1))$.
Let $P_1$ be the first point, that is 'P1=E(13308581970510900443a + 15361864644717267429, 12035351063104026383a + 6630581890190451124)' and use the code 'P1.order().factor()' to get the order of $P_1$, which is $2^2 * 4999 * 922521708027083$. Now, let 'Q1=2^2 * 922521708027083 * P1', then $Q_1$ is of the required order. We can repeat this with the other point.
| | 4 | No.4 Revision |
An answer could be as follows:
'p=18446744073709551667'
then define the field
'K . < a >=GF(p^2)'
over which we define the curve
'E = EllipticCurve(K,[1,0])'. Now use the code
'E.cardinality().factor()' to get the cardinality of $E$, which is $2^4 * 4999^2 * 922521708027083^2$.
Use the code 'pts = E.gens()' to find the generators which turn out to be $((3967812587828368975${(3967812587828368975a + 7437411822947458617 : 16672546567636985011a + 14299206839437184441 : 1)$, $ 1), (8681109523785822829a + 7072963597633280041 : 15515941078240688329a + 11174851755365315891 : 1))$. 1)}$.
Let $P_1$ be the first point, that is 'P1=E(13308581970510900443a + 15361864644717267429, 12035351063104026383a + 6630581890190451124)' and use the code 'P1.order().factor()' to get the order of $P_1$, which is $2^2 * 4999 * 922521708027083$. Now, let 'Q1=2^2 * 922521708027083 * P1', then $Q_1$ is of the required order. We can repeat this with the other point.
| | 5 | No.5 Revision |
An answer could be as follows: 'p=18446744073709551667' then define the field 'K . < a >=GF(p^2)' over which we define the curve 'E = EllipticCurve(K,[1,0])'. Now use the code 'E.cardinality().factor()' to get the cardinality of $E$, which is $2^4 * 4999^2 * 922521708027083^2$. Use the code 'pts = E.gens()' to find the generators which turn out to be ${(3967812587828368975a + 7437411822947458617 : 16672546567636985011a + 14299206839437184441 : 1), (8681109523785822829a + 7072963597633280041 : 15515941078240688329a + 11174851755365315891 : 1)}$.
Let $P_1$ be the first point, that is 'P1=E(13308581970510900443a + 15361864644717267429, 12035351063104026383a + 6630581890190451124)' and use the code 'P1.order().factor()' to get the order of $P_1$, which is $2^2 * 4999 * 922521708027083$. Now, let 'Q1=2^2 * 922521708027083 * P1', then $Q_1$ is of the required order. We can repeat this with the other point.
