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I don't understand How to do Extension field
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$EllipticCurve:y^2=x^3+x+300 pmod8111 $ has order $8269$
This curve on $mathbbF_8111$
If expand this equation to $mathbbF_8111^3$
The elliptic curve is defined as : $y^2=x^3+x+300 pmod8111^3 $ and Polynomial : $x^3+4x-11$
On the SAGECODE :
F. = GF(8111)
K. = GF(8111^3, name='x', modulus=x^3 + 4*x - 11)
E = EllipticCurve(GF(8111), [1,300])
E_ = EllipticCurve(K, [1,300])
The questions are:
How did you choose $mathbbF_8111^3$ Polynomial??
And how do I expand to $mathbbF_8111^2$
.
I did not speak English well and I used a translator. I am really sorry if the question was rude.
thanks you for read
finite-fields extension-field
asked 2 days ago
byks
82
add a comment |Â
up vote
0
down vote
favorite
$EllipticCurve:y^2=x^3+x+300 pmod8111 $ has order $8269$
This curve on $mathbbF_8111$
If expand this equation to $mathbbF_8111^3$
The elliptic curve is defined as : $y^2=x^3+x+300 pmod8111^3 $ and Polynomial : $x^3+4x-11$
On the SAGECODE :
F. = GF(8111)
K. = GF(8111^3, name='x', modulus=x^3 + 4*x - 11)
E = EllipticCurve(GF(8111), [1,300])
E_ = EllipticCurve(K, [1,300])
The questions are:
How did you choose $mathbbF_8111^3$ Polynomial??
And how do I expand to $mathbbF_8111^2$
.
I did not speak English well and I used a translator. I am really sorry if the question was rude.
thanks you for read
finite-fields extension-field
asked 2 days ago
byks
82
1
The field with $8111^3$ elements is not arithmetic modulo $8111^3$. You need to find a cubic irreducible polynomial over $mathbb F_8111$ and do arithmetic modulo that.
– Henning Makholm
2 days ago
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
$EllipticCurve:y^2=x^3+x+300 pmod8111 $ has order $8269$
This curve on $mathbbF_8111$
If expand this equation to $mathbbF_8111^3$
The elliptic curve is defined as : $y^2=x^3+x+300 pmod8111^3 $ and Polynomial : $x^3+4x-11$
On the SAGECODE :
F. = GF(8111)
K. = GF(8111^3, name='x', modulus=x^3 + 4*x - 11)
E = EllipticCurve(GF(8111), [1,300])
E_ = EllipticCurve(K, [1,300])
The questions are:
How did you choose $mathbbF_8111^3$ Polynomial??
And how do I expand to $mathbbF_8111^2$
.
I did not speak English well and I used a translator. I am really sorry if the question was rude.
thanks you for read
finite-fields extension-field
asked 2 days ago
byks
82
$EllipticCurve:y^2=x^3+x+300 pmod8111 $ has order $8269$
This curve on $mathbbF_8111$
If expand this equation to $mathbbF_8111^3$
The elliptic curve is defined as : $y^2=x^3+x+300 pmod8111^3 $ and Polynomial : $x^3+4x-11$
On the SAGECODE :
F. = GF(8111)
K. = GF(8111^3, name='x', modulus=x^3 + 4*x - 11)
E = EllipticCurve(GF(8111), [1,300])
E_ = EllipticCurve(K, [1,300])
The questions are:
How did you choose $mathbbF_8111^3$ Polynomial??
And how do I expand to $mathbbF_8111^2$
.
I did not speak English well and I used a translator. I am really sorry if the question was rude.
thanks you for read
finite-fields extension-field
asked 2 days ago
byks
82
asked 2 days ago
byks
82
asked 2 days ago
byks
82
asked 2 days ago
byks
82
82
1
The field with $8111^3$ elements is not arithmetic modulo $8111^3$. You need to find a cubic irreducible polynomial over $mathbb F_8111$ and do arithmetic modulo that.
– Henning Makholm
2 days ago
add a comment |Â
1
The field with $8111^3$ elements is not arithmetic modulo $8111^3$. You need to find a cubic irreducible polynomial over $mathbb F_8111$ and do arithmetic modulo that.
– Henning Makholm
2 days ago
1
1
The field with $8111^3$ elements is not arithmetic modulo $8111^3$. You need to find a cubic irreducible polynomial over $mathbb F_8111$ and do arithmetic modulo that.
– Henning Makholm
2 days ago
The field with $8111^3$ elements is not arithmetic modulo $8111^3$. You need to find a cubic irreducible polynomial over $mathbb F_8111$ and do arithmetic modulo that.
– Henning Makholm
2 days ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
0
down vote
accepted
Let us do things in sage then.
The elliptic curve is
sage: p = 8111
sage: p.is_prime()
True
sage: F = GF(p)
sage: E = EllipticCurve( F, [1, 300] )
sage: E.order().factor()
8269
sage: K = GF(p^3)
sage: K.modulus()
x^3 + 4*x + 8100
sage: # the above is x^3 + 4x - 11 mod 8111
sage: # we can count points over K...
sage: E.base_extend(K).order().factor()
2^2 * 1951 * 8269^2
sage: N = E.order() # later edit, forgotten first, sorry
sage: N
8269
sage: a,b = (x^2 - (p+1-N)*x + p).roots(ring=QQbar, multiplicities=False)
sage: ZZ(p^3 -a^3 - b^3 + 1)
533609121244
sage: _.factor()
2^2 * 1951 * 8269^2
Note that sage is choosing the same modulus. If we want to specify explicitly the modulus of the field extension $Bbb F_p^3$, $p=8111$, then we can do this as follows:
sage: R.<X> = PolynomialRing(F)
sage: K.<a> = GF(p^3, modulus = X^3 + 4*X - 11)
sage: K
Finite Field in a of size 8111^3
sage: K.modulus()
x^3 + 4*x + 8100
sage: a.minpoly()
x^3 + 4*x + 8100
The curve $E/K$ can be introduced either directly, or after a base extension, E.base_extend(K),
sage: EK = EllipticCurve( K, [1, 300] )
sage: EK == E.base_extend(K)
True
A very good sage support is provided by asksage.
answered 2 days ago
dan_fulea
3,8921211
Oh my god , Thanks you so much!
– byks
2 days ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
Let us do things in sage then.
The elliptic curve is
sage: p = 8111
sage: p.is_prime()
True
sage: F = GF(p)
sage: E = EllipticCurve( F, [1, 300] )
sage: E.order().factor()
8269
sage: K = GF(p^3)
sage: K.modulus()
x^3 + 4*x + 8100
sage: # the above is x^3 + 4x - 11 mod 8111
sage: # we can count points over K...
sage: E.base_extend(K).order().factor()
2^2 * 1951 * 8269^2
sage: N = E.order() # later edit, forgotten first, sorry
sage: N
8269
sage: a,b = (x^2 - (p+1-N)*x + p).roots(ring=QQbar, multiplicities=False)
sage: ZZ(p^3 -a^3 - b^3 + 1)
533609121244
sage: _.factor()
2^2 * 1951 * 8269^2
Note that sage is choosing the same modulus. If we want to specify explicitly the modulus of the field extension $Bbb F_p^3$, $p=8111$, then we can do this as follows:
sage: R.<X> = PolynomialRing(F)
sage: K.<a> = GF(p^3, modulus = X^3 + 4*X - 11)
sage: K
Finite Field in a of size 8111^3
sage: K.modulus()
x^3 + 4*x + 8100
sage: a.minpoly()
x^3 + 4*x + 8100
The curve $E/K$ can be introduced either directly, or after a base extension, E.base_extend(K),
sage: EK = EllipticCurve( K, [1, 300] )
sage: EK == E.base_extend(K)
True
A very good sage support is provided by asksage.
answered 2 days ago
dan_fulea
3,8921211
Oh my god , Thanks you so much!
– byks
2 days ago
add a comment |Â
up vote
0
down vote
accepted
Let us do things in sage then.
The elliptic curve is
sage: p = 8111
sage: p.is_prime()
True
sage: F = GF(p)
sage: E = EllipticCurve( F, [1, 300] )
sage: E.order().factor()
8269
sage: K = GF(p^3)
sage: K.modulus()
x^3 + 4*x + 8100
sage: # the above is x^3 + 4x - 11 mod 8111
sage: # we can count points over K...
sage: E.base_extend(K).order().factor()
2^2 * 1951 * 8269^2
sage: N = E.order() # later edit, forgotten first, sorry
sage: N
8269
sage: a,b = (x^2 - (p+1-N)*x + p).roots(ring=QQbar, multiplicities=False)
sage: ZZ(p^3 -a^3 - b^3 + 1)
533609121244
sage: _.factor()
2^2 * 1951 * 8269^2
Note that sage is choosing the same modulus. If we want to specify explicitly the modulus of the field extension $Bbb F_p^3$, $p=8111$, then we can do this as follows:
sage: R.<X> = PolynomialRing(F)
sage: K.<a> = GF(p^3, modulus = X^3 + 4*X - 11)
sage: K
Finite Field in a of size 8111^3
sage: K.modulus()
x^3 + 4*x + 8100
sage: a.minpoly()
x^3 + 4*x + 8100
The curve $E/K$ can be introduced either directly, or after a base extension, E.base_extend(K),
sage: EK = EllipticCurve( K, [1, 300] )
sage: EK == E.base_extend(K)
True
A very good sage support is provided by asksage.
answered 2 days ago
dan_fulea
3,8921211
Oh my god , Thanks you so much!
– byks
2 days ago
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
Let us do things in sage then.
The elliptic curve is
sage: p = 8111
sage: p.is_prime()
True
sage: F = GF(p)
sage: E = EllipticCurve( F, [1, 300] )
sage: E.order().factor()
8269
sage: K = GF(p^3)
sage: K.modulus()
x^3 + 4*x + 8100
sage: # the above is x^3 + 4x - 11 mod 8111
sage: # we can count points over K...
sage: E.base_extend(K).order().factor()
2^2 * 1951 * 8269^2
sage: N = E.order() # later edit, forgotten first, sorry
sage: N
8269
sage: a,b = (x^2 - (p+1-N)*x + p).roots(ring=QQbar, multiplicities=False)
sage: ZZ(p^3 -a^3 - b^3 + 1)
533609121244
sage: _.factor()
2^2 * 1951 * 8269^2
Note that sage is choosing the same modulus. If we want to specify explicitly the modulus of the field extension $Bbb F_p^3$, $p=8111$, then we can do this as follows:
sage: R.<X> = PolynomialRing(F)
sage: K.<a> = GF(p^3, modulus = X^3 + 4*X - 11)
sage: K
Finite Field in a of size 8111^3
sage: K.modulus()
x^3 + 4*x + 8100
sage: a.minpoly()
x^3 + 4*x + 8100
The curve $E/K$ can be introduced either directly, or after a base extension, E.base_extend(K),
sage: EK = EllipticCurve( K, [1, 300] )
sage: EK == E.base_extend(K)
True
A very good sage support is provided by asksage.
answered 2 days ago
dan_fulea
3,8921211
Let us do things in sage then.
The elliptic curve is
sage: p = 8111
sage: p.is_prime()
True
sage: F = GF(p)
sage: E = EllipticCurve( F, [1, 300] )
sage: E.order().factor()
8269
sage: K = GF(p^3)
sage: K.modulus()
x^3 + 4*x + 8100
sage: # the above is x^3 + 4x - 11 mod 8111
sage: # we can count points over K...
sage: E.base_extend(K).order().factor()
2^2 * 1951 * 8269^2
sage: N = E.order() # later edit, forgotten first, sorry
sage: N
8269
sage: a,b = (x^2 - (p+1-N)*x + p).roots(ring=QQbar, multiplicities=False)
sage: ZZ(p^3 -a^3 - b^3 + 1)
533609121244
sage: _.factor()
2^2 * 1951 * 8269^2
Note that sage is choosing the same modulus. If we want to specify explicitly the modulus of the field extension $Bbb F_p^3$, $p=8111$, then we can do this as follows:
sage: R.<X> = PolynomialRing(F)
sage: K.<a> = GF(p^3, modulus = X^3 + 4*X - 11)
sage: K
Finite Field in a of size 8111^3
sage: K.modulus()
x^3 + 4*x + 8100
sage: a.minpoly()
x^3 + 4*x + 8100
The curve $E/K$ can be introduced either directly, or after a base extension, E.base_extend(K),
sage: EK = EllipticCurve( K, [1, 300] )
sage: EK == E.base_extend(K)
True
A very good sage support is provided by asksage.
answered 2 days ago
dan_fulea
3,8921211
edited yesterday
answered 2 days ago
dan_fulea
3,8921211
answered 2 days ago
dan_fulea
3,8921211
answered 2 days ago
dan_fulea
3,8921211
3,8921211
Oh my god , Thanks you so much!
– byks
2 days ago
add a comment |Â
Oh my god , Thanks you so much!
– byks
2 days ago
Oh my god , Thanks you so much!
– byks
2 days ago
Oh my god , Thanks you so much!
– byks
2 days ago
add a comment |Â
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1
The field with $8111^3$ elements is not arithmetic modulo $8111^3$. You need to find a cubic irreducible polynomial over $mathbb F_8111$ and do arithmetic modulo that.
– Henning Makholm
2 days ago