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Find the Order of an Elliptic Curve

Find the rational points on $1 + 18 x + 81 x^2 + 44 x^3 = y^2$ with SageSage usage to calculate a cardinalityProblem with Elliptic Curve in Montgomery formElliptic curve Schoof algorithm, projective polynomial point coordinatesFind all points of finite order on the elliptic curve $y^2+7xy=x^3+16x$.How do you compute order of points in elliptic curve?Given order of two points, determining the number of points on an elliptic curveManually Finding Points Along Elliptic CurveCalculus of order in an elliptic curveWhy are zeroes of the elliptic curve (mod p) for integer values symmetrical about p/2

0

$begingroup$

I have an Elliptic Curve represented by the following equation and values:

Elliptic Curve: y^2 = x^3 + A*x + B mod M

M = 93556643250795678718734474880013829509320385402690660619699653921022012489089
A = 66001598144012865876674115570268990806314506711104521036747533612798434904785
B = 25255205054024371783896605039267101837972419055969636393425590261926131199030

What is the order of this Elliptic curve?

I thought it should be the same as the modulo, M?

In sage math, if I check the following, I get a different result:

F = FiniteField(M)
E = EllipticCurve(F, [A, B])

E.order() = 93556643250795678718734474880013829509196181230338248789325711173791286325820

However, the value of M is:
93556643250795678718734474880013829509320385402690660619699653921022012489089

These values look similar but they are not the same.

So, I want to know the following:

How is the order of the Elliptic Curve calculated?
How is it related to modulus? Since there is a similarity between the two values?

Thanks.

elliptic-curves sagemath

share|cite|improve this question

asked Mar 29 at 3:01

Neon FlashNeon Flash

1012

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The order of an elliptic curve is the number $N$ of points on it (including the "point at infinity"). If it is defined modulo $p$, then $|N-p-1|le2sqrt p$ (Hasse). I would expect Sage uses some form of Schoof's algorithm.
  $endgroup$
  – Lord Shark the Unknown
  Mar 29 at 3:11
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The order of an elliptic curve over finite field need not be equal to the order of the field. It might be instructive to calculate the number of solutions mod p for the equation $y=x(x^2-1) pmod p$ for small values of $p= 5,7,11$ (they are all elliptic curves) and see for yourself. You can see Theorem 1.1, Ch. V in SIlverman's book "Arithmetic of Elliptic Curves".
  $endgroup$
  – P Vanchinathan
  Mar 29 at 3:14
  

  
  
  
  

add a comment | 

0

$begingroup$

I have an Elliptic Curve represented by the following equation and values:

Elliptic Curve: y^2 = x^3 + A*x + B mod M

M = 93556643250795678718734474880013829509320385402690660619699653921022012489089
A = 66001598144012865876674115570268990806314506711104521036747533612798434904785
B = 25255205054024371783896605039267101837972419055969636393425590261926131199030

What is the order of this Elliptic curve?

I thought it should be the same as the modulo, M?

In sage math, if I check the following, I get a different result:

F = FiniteField(M)
E = EllipticCurve(F, [A, B])

E.order() = 93556643250795678718734474880013829509196181230338248789325711173791286325820

However, the value of M is:
93556643250795678718734474880013829509320385402690660619699653921022012489089

These values look similar but they are not the same.

So, I want to know the following:

How is the order of the Elliptic Curve calculated?
How is it related to modulus? Since there is a similarity between the two values?

Thanks.

elliptic-curves sagemath

share|cite|improve this question

asked Mar 29 at 3:01

Neon FlashNeon Flash

1012

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  The order of an elliptic curve is the number $N$ of points on it (including the "point at infinity"). If it is defined modulo $p$, then $|N-p-1|le2sqrt p$ (Hasse). I would expect Sage uses some form of Schoof's algorithm.
  $endgroup$
  – Lord Shark the Unknown
  Mar 29 at 3:11
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The order of an elliptic curve over finite field need not be equal to the order of the field. It might be instructive to calculate the number of solutions mod p for the equation $y=x(x^2-1) pmod p$ for small values of $p= 5,7,11$ (they are all elliptic curves) and see for yourself. You can see Theorem 1.1, Ch. V in SIlverman's book "Arithmetic of Elliptic Curves".
  $endgroup$
  – P Vanchinathan
  Mar 29 at 3:14
  

  
  
  
  

add a comment | 

$begingroup$

I have an Elliptic Curve represented by the following equation and values:

Elliptic Curve: y^2 = x^3 + A*x + B mod M

M = 93556643250795678718734474880013829509320385402690660619699653921022012489089
A = 66001598144012865876674115570268990806314506711104521036747533612798434904785
B = 25255205054024371783896605039267101837972419055969636393425590261926131199030

What is the order of this Elliptic curve?

I thought it should be the same as the modulo, M?

In sage math, if I check the following, I get a different result:

F = FiniteField(M)
E = EllipticCurve(F, [A, B])

E.order() = 93556643250795678718734474880013829509196181230338248789325711173791286325820

However, the value of M is:
93556643250795678718734474880013829509320385402690660619699653921022012489089

These values look similar but they are not the same.

So, I want to know the following:

How is the order of the Elliptic Curve calculated?
How is it related to modulus? Since there is a similarity between the two values?

Thanks.

elliptic-curves sagemath

share|cite|improve this question

asked Mar 29 at 3:01

Neon FlashNeon Flash

1012

$endgroup$

M = 93556643250795678718734474880013829509320385402690660619699653921022012489089
A = 66001598144012865876674115570268990806314506711104521036747533612798434904785
B = 25255205054024371783896605039267101837972419055969636393425590261926131199030

F = FiniteField(M)
E = EllipticCurve(F, [A, B])

E.order() = 93556643250795678718734474880013829509196181230338248789325711173791286325820

share|cite|improve this question

asked Mar 29 at 3:01

Neon FlashNeon Flash

1012

share|cite|improve this question

asked Mar 29 at 3:01

Neon FlashNeon Flash

1012

asked Mar 29 at 3:01

Neon FlashNeon Flash

1012

asked Mar 29 at 3:01

Neon FlashNeon Flash

1012