points on elliptic cuver over 2^m field

A is a random element

A=a^107 + a^105 + a^104 + a^100 + a^99 + a^98 + a^96 + a^94 + a^93 + a^83 + a^75 + a^74 + a^71 + a^70 + a^68 + a^67 + a^66 + a^64 + a^63 + a^60 + a^58 + a^56 + a^54 + a^53 + a^52 + a^50 + a^49 + a^46 + a^44 + a^42 + a^41 + a^39 + a^37 + a^35 + a^34 + a^33 + a^31 + a^30 + a^29 + a^23 + a^22 + a^21 + a^20 + a^16 + a^13 + a^12 + a^9 + a^8 + a^6 + a^3 + a^2 + a + 1

B is random element

B=a^112 + a^109 + a^108 + a^106 + a^105 + a^104 + a^103 + a^102 + a^101 + a^100 + a^97 + a^96 + a^93 + a^90 + a^87 + a^84 + a^81 + a^79 + a^78 + a^77 + a^76 + a^75 + a^71 + a^70 + a^69 + a^68 + a^67 + a^65 + a^63 + a^59 + a^55 + a^54 + a^53 + a^52 + a^51 + a^50 + a^48 + a^47 + a^46 + a^44 + a^43 + a^42 + a^41 + a^39 + a^38 + a^37 + a^36 + a^35 + a^32 + a^30 + a^28 + a^27 + a^23 + a^22 + a^21 + a^16 + a^15 + a^14 + a^12 + a^11 + a^9 + a^7 + a^4 + a + 1

E = EllipticCurve( F, [ 1, A, 0, 0, B ]);E print (E.points()[:2])

when i run the above code it is difficult to print point on the curve as it is required lot of time. how to generate point?

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Please reformat the own post, so that the code is marked as such. The enter code here shows the button with 101and 010 was pressed, but not "on the code". To get it work one can

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dan_fulea ( 2018-01-06 14:57:40 +0100 )edit

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answered 2018-01-06 15:07:52 +0100

dan_fulea
5685 ●5 ●44 ●97

Short answer:

The needed method to get one random point on the constructed elliptic curve is E.random_point().

Longer answer:

First of all, in questions and answers, it is better to use a situation that can be reconstructed, if possible. This method E.random_point()is itself unpredictable, but the curve E will be at least fixed.

The following code

F.<a> = GF( 2^113 )
A, B = a+2017, a+2018
E = EllipticCurve( F, [ 1, A, 0, 0, B ] )
E.random_point()

produces (this time):

(a^112 + a^108 + a^105 + a^104 + a^102 + a^98 + a^96 + a^95 + a^94 + a^93 + a^91 + a^90 + a^89 + a^85 + a^83 + a^82 + a^81 + a^76 + a^74 + a^73 + a^69 + a^67 + a^66 + a^65 + a^62 + a^61 + a^60 + a^55 + a^54 + a^52 + a^51 + a^49 + a^48 + a^45 + a^39 + a^37 + a^33 + a^32 + a^31 + a^30 + a^28 + a^24 + a^23 + a^21 + a^19 + a^18 + a^15 + a^13 + a^10 + a^6 + a^5 + a^2 + a + 1 : a^112 + a^107 + a^105 + a^103 + a^101 + a^100 + a^98 + a^97 + a^95 + a^94 + a^93 + a^92 + a^90 + a^89 + a^88 + a^87 + a^86 + a^85 + a^82 + a^81 + a^80 + a^79 + a^74 + a^72 + a^71 + a^70 + a^69 + a^67 + a^66 + a^65 + a^61 + a^56 + a^55 + a^54 + a^52 + a^51 + a^50 + a^48 + a^47 + a^45 + a^44 + a^43 + a^42 + a^40 + a^39 + a^36 + a^35 + a^34 + a^33 + a^32 + a^29 + a^27 + a^25 + a^24 + a^19 + a^17 + a^16 + a^15 + a^13 + a^12 + a^7 + a^5 + a^3 + a : 1)

Note that computing all points (and then taking the first two) is not a good idea, since there are too many, and they must all be (computed and) stored, before the first two are shown. And also, these first two points are not really random, for instance:

sage: EllipticCurve( GF(11), [4,5] ).points()[:2]
[(0 : 1 : 0), (0 : 4 : 1)]

and the first point is $O$.

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points on elliptic cuver over 2^m field

A is a random element

B is random element

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points on elliptic cuver over 2^m field edit

A is a random element

B is random element

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points on elliptic cuver over 2^m field