ECC Result Verification

From: flip (flip_alpha_at_safebunch.com)
Date: 11/02/04 

Date: Mon, 1 Nov 2004 18:34:31 -0800

Hi All,

can someone please verify these results?

E: y^2 = x^3 + 1 x + 1 (mod 23)

The set of points that satisfy E are given by:

{{0,1}, {0,22}, {1,7}, {1,16}, {3,10}, {3,13}, {4,0}, {5,4}, {5,19}, {6,4},
{6,19}, {7,11}, {7,12}, {9,7}, {9,16}, {11,3}, {11,20}, {12,4}, {12,19},
{13,7}, {13,16}, {17,3}, {17,20}, {18,3}, {18,20}, {19,5}, {19,18}}

There are a total of 28 points, so # E(F23) = 28

The order of each point is {28, 28, 28, 28, 28, 28, 2, 7, 7, 14, 14, 14, 14,
28, 28, 4, 4, 14, 14, 7, 7, 7, 7, 28, 28, 28, 28}

The cofactor (h) of each point is {1, 1, 1, 1, 1, 1, 14, 4, 4, 2, 2, 2, 2,
1, 1, 7, 7, 2, 2, 4, 4, 4, 4, 1, 1, 1, 1}

Choose the point {0, 1} as a generator point and get:

1P = (0, 1) 11P = (1, 16) 21P = (11, 20)
2P = (6, 19) 12P = (17, 20) 22P = (7, 12)
3P = (3, 13) 13P = (9, 16) 23P = (18, 20)
4P = (13, 16) 14P = (4, 0) 24P = (13, 7)
5P = (18, 3) 15P = (9, 7) 25P = (3, 10)
6P = (7, 11) 16P = (17, 3) 26P = (6, 4)
7P = (11, 3) 17P = (1, 7) 27P = (0, 22)
8P = (5, 19) 18P = (12, 19) 28P = (0, 1) = 1P (point at infinity)
9P = (19, 18) 19P = (19, 5)
10P = (12, 4) 20P = (5, 4)

Lastly: 17*(19, 18) and 9*(1, 7) = (9, 16) (this did not match the book,
which I think might be wrong).

Thanks!

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