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EllipticCurves :: bruteForceRationalPoints(EllipticCurveW)

bruteForceRationalPoints(EllipticCurveW) -- A brute-force method to list the points on an elliptic curve.

Synopsis

Description

i1 : E=ellCurve({1,2,3,4,6},GF(29))

o1 = Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).

o1 : EllipticCurveW
i2 : bruteForceRationalPoints(E)

o2 = {Point  {0, 1, 0}  on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y =
     ------------------------------------------------------------------------
     X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {2, 12, 1}  on the
     ------------------------------------------------------------------------
     (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6
     ------------------------------------------------------------------------
     over (GF 29).), Point  {4, -12, 1}  on the (Elliptic Curve defined by
     ------------------------------------------------------------------------
     Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {4, 5,
     ------------------------------------------------------------------------
     1}  on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4
     ------------------------------------------------------------------------
     X + 6  over (GF 29).), Point  {8, -2, 1}  on the (Elliptic Curve defined
     ------------------------------------------------------------------------
     by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {8,
     ------------------------------------------------------------------------
     -9, 1}  on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2
     ------------------------------------------------------------------------
     + 4 X + 6  over (GF 29).), Point  {-13, -13, 1}  on the (Elliptic Curve
     ------------------------------------------------------------------------
     defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).),
     ------------------------------------------------------------------------
     Point  {-13, -6, 1}  on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y
     ------------------------------------------------------------------------
     = X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {-5, -11, 1}  on the
     ------------------------------------------------------------------------
     (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6
     ------------------------------------------------------------------------
     over (GF 29).), Point  {-5, 13, 1}  on the (Elliptic Curve defined by
     ------------------------------------------------------------------------
     Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {-11,
     ------------------------------------------------------------------------
     -3, 1}  on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2
     ------------------------------------------------------------------------
     + 4 X + 6  over (GF 29).), Point  {-11, 11, 1}  on the (Elliptic Curve
     ------------------------------------------------------------------------
     defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).),
     ------------------------------------------------------------------------
     Point  {7, 1, 1}  on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y =
     ------------------------------------------------------------------------
     X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {7, -11, 1}  on the
     ------------------------------------------------------------------------
     (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6
     ------------------------------------------------------------------------
     over (GF 29).), Point  {-1, 1, 1}  on the (Elliptic Curve defined by Y^2
     ------------------------------------------------------------------------
     + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {-1, -3, 1}
     ------------------------------------------------------------------------
     on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X +
     ------------------------------------------------------------------------
     6  over (GF 29).), Point  {-2, 7, 1}  on the (Elliptic Curve defined by
     ------------------------------------------------------------------------
     Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {-2,
     ------------------------------------------------------------------------
     -8, 1}  on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2
     ------------------------------------------------------------------------
     + 4 X + 6  over (GF 29).), Point  {-4, 12, 1}  on the (Elliptic Curve
     ------------------------------------------------------------------------
     defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).),
     ------------------------------------------------------------------------
     Point  {-4, -11, 1}  on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y
     ------------------------------------------------------------------------
     = X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {-8, 1, 1}  on the
     ------------------------------------------------------------------------
     (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6
     ------------------------------------------------------------------------
     over (GF 29).), Point  {-8, 4, 1}  on the (Elliptic Curve defined by Y^2
     ------------------------------------------------------------------------
     + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {-6, 10, 1}
     ------------------------------------------------------------------------
     on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X +
     ------------------------------------------------------------------------
     6  over (GF 29).), Point  {-6, -7, 1}  on the (Elliptic Curve defined by
     ------------------------------------------------------------------------
     Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {10,
     ------------------------------------------------------------------------
     -10, 1}  on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2
     ------------------------------------------------------------------------
     X^2 + 4 X + 6  over (GF 29).), Point  {10, -3, 1}  on the (Elliptic
     ------------------------------------------------------------------------
     Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF
     ------------------------------------------------------------------------
     29).), Point  {-14, 8, 1}  on the (Elliptic Curve defined by Y^2 + 1 XY
     ------------------------------------------------------------------------
     + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {-14, 3, 1}  on
     ------------------------------------------------------------------------
     the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6
     ------------------------------------------------------------------------
     over (GF 29).), Point  {0, 12, 1}  on the (Elliptic Curve defined by Y^2
     ------------------------------------------------------------------------
     + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X + 6  over (GF 29).), Point  {0, 14, 1}
     ------------------------------------------------------------------------
     on the (Elliptic Curve defined by Y^2 + 1 XY + 3 Y = X^3 + 2 X^2 + 4 X +
     ------------------------------------------------------------------------
     6  over (GF 29).)}

o2 : List