I am trying to add points on an elliptic curve and failing. (Using formulas from William and Tate Rational Points of Elliptic Curves.)
$$ C: \; y^2 = x^3 + x^2 +x + 3 \quad \text{over the field} \quad \mathbb{F}_{103}. $$
$P = (7,14)$
$a=1, b=1, c=3$ (the general formula for an elliptic curve is $y^2 = x^3 +ax^2+bx +c$).
all calculations done modulo 103
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$$2P$$
$$ \lambda = \frac{ 3x^2+2ax+b}{2y} = \frac{ 162}{28} = (162)(92) = 72 $$
$$ \nu = y-\lambda x = 14 – 72(7) = 25 $$
$$ x(2P) = \lambda ^2 -a-2x = 72^2-1-2(7) = 19 $$
$$ y(2P) = \lambda \:( x(2P)) + \nu = 72(19) + 25 = 54$$
$$ \Rightarrow 2P = (19,54) .$$
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Now using the duplication formula:
$$2P$$
$$x(2P) = \frac{x^4 – 2bx^2 – 8cx + b^2 – 4ac}{4x^3 + 4ax^2 + 4bx + 4c} = \frac{2460}{1608} = 2460\cdot 18 = 93$$
$$\lambda = \frac{3x^2+2ax+b}{2y} = \frac{ 162}{28} = (162)(92) = 72 $$
$$v=y-\lambda x = 14 – (72)7 =25 $$
$$ y(2P) = \lambda x(2P) + \nu = 72(93)+25 = 26$$
$$ \Rightarrow 2P = (93,26) $$
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$$ (93,26) \neq (19,54) \Rightarrow 2P \neq 2P.$$
Where did I go wrong?
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I used these formulas
$$\lambda = \frac{y_2-y_1}{x_2-x_1} $$
$$ x(nP) = \lambda ^2 -1-x_1-x_2 $$
$$ v=y_1-\lambda x_1 $$
$$ y(nP) = \lambda \: (x(nP)) + \nu $$
to continue calculations for $nP = (n-1)P +P$, $n$ an integer, but I got $2P = 4P$, are these the correct formulas to be using?
Best Answer
For the numerator of your duplication formula we have $$7^4-2\cdot 7^2-8\cdot 3\cdot 7+1-4\cdot 3=2124$$ Then $2124\cdot 18=19$ modulo $103$, so the formulae agree.