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I have calculated the affine points on the curve $x^2 + y^2 = 1 − 3x^2y^2$ over the field ${\mathbb{Z}}_{11}.$

Using $y^2 = \frac{1-x^2}{1+3x^2}$ I got the following points:

$(0,1),(0,10),(1,0),(2,2),(2,9),(9,2),(9,9),(10,0)$

Total comes to be 8 points. But what I know is that total points should come 11-1 = 10.

So I am not getting other 2 points.

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    $\begingroup$ How do you conclude that the number of points must be 11-1 ? $\endgroup$ – user27950 Oct 25 '15 at 12:18
  • $\begingroup$ Are you sure about the $x^2y^2$ part? In any case, this is probably off-topic. $\endgroup$ – fkraiem Oct 25 '15 at 13:37
  • $\begingroup$ I read it some where that the order normally is $p-1$ but I am not sure. Also here this curve is over $\mathbb{F}_{11}$, so I tried using 11-1 = 10 $\endgroup$ – TechJ Oct 25 '15 at 14:11
  • $\begingroup$ I think you confuse that with the order of $GF(p)^*$, which is $p-1$. The order of an elliptic curve cannot be calculated so easily. The curve of your question has exactely the 8 points you stated. $\endgroup$ – user27950 Oct 25 '15 at 14:42
  • $\begingroup$ This is an Edwards curve, right? Are you looking for the points over the integers, or over some finite field? $\endgroup$ – Circonflexe Oct 30 '15 at 9:18
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Well this curve has 8 affine points and I would like to share the way to find them all.

First of all we make square table till 10 as we have field ${\mathbb{Z}}_{11}.$ (so 11-1=10)

Below is the table(also calculate its mod 11 values) :

  ┌───────┬─────┬────┬────┬────┬────┬────┬────┬────┬─────┬─────┬────┬
  │   #   │  0  │  1 │  2 │  3 │ 4  │ 5  │ 6  │  7 │  8  │  9  │ 10 │
  └───────┴─────┴────┴────┴────┴────┴────┴────┴────┴─────┴─────┴────┴    
  │Square │  0  │  1 │  4 │  9 │ 16 │ 25 │ 36 │ 49 │  64 │  81 │ 100│
  └───────┴─────┴────┴────┴────┴────┴────┴────┴────┴─────┴─────┴────┴    
  │Mod 11 │  0  │  1 │  4 │  9 │ 5  │ 3  │ 3  │  5 │  9  │  4  │  1 │
  └───────┴─────┴────┴────┴────┴────┴────┴────┴────┴─────┴─────┴────┴    

Re-arranging the equation so that we have $y$ on left hand side

$y^2 = \frac{1-x^2}{1+3x^2}$

Now we start putting form $x = 0$ until $x = 10$ in the above equation and get the value of $y$ each time. If value of $y$ is one of the complete squares mod 11 i.e row 3rd of the table then corresponding value in row 1 is the $y$ co-ordinate.

$y$ is calculated as mod 11. Even if the value of $y$ is in fractions.

For $x=0$ we get $y=1$ which exists in row 3rd i.e it is a complete square. Now value 1 in 3rd row corresponds to value $1$ and $10$ in row 1. So we get 2 points here i.e. $(0,1)$ and $(0,10)$

For $x=1$ we get $y=0$ this also exists in 3rd row. So corresponding values in 1st row are 0. So here we have only 1 point $(1,0)$

For $x=2$ we get $y=4$ this also exists in 3rd row. So corresponding values in 1st row are 2 and 9. So here we have 2 points i.e. $(2,2)$ and $(2,9)$

For $x=3$ we get $y=6$ which does not exists in the 3rd row of the table. So no corresponding value of $y$ will be there, hence no point will exist at $x=3$

For $x=4$ we get $y=8$ Not in table (not a complete square) hence no points here.

For $x=5$ we get $y=2$ Not in table (not a complete square) hence no points here.

For $x=6$ we get $y=2$ Not in table (not a complete square) hence no points here.

For $x=7$ we get $y=8$ Not in table (not a complete square) hence no points here.

For $x=8$ we get $y=6$ Not in table (not a complete square) hence no points here.

For $x=9$ we get $y=4$ Exists in table (hence it is a complete square) So corresponding points are: $(9,2)$ and $(9,9)$

For $x=10$ we get $y=0$ Exists in table (hence it is a complete square) So corresponding point is: $(10,0)$

So all the affine points are $(0,1)$ $(0,10)$ $(1,0)$ $(2,2)$$(2,9)$ $(9,2)$ $(9,9)$ $(10,0)$

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