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Fix whitespace characters

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Henry de Valence 2018-04-04 17:40:37 -07:00
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docs/ristretto-notes.md
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@ -75,19 +75,19 @@ to be square), the curve is *complete*. In this case the
four-torsion subgroup is cyclic, and we four-torsion subgroup is cyclic, and we
can write it explicitly as can write it explicitly as
$$ $$
\mathcal E\_{a,d}[4] = \\{ (0,1),\; (1/\sqrt a, 0),\; (0, -1),\; (-1/\sqrt{a}, 0)\\}. \mathcal E\_{a,d}[4] = \\{ (0,1),\\; (1/\sqrt a, 0),\\; (0, -1),\\; (-1/\sqrt{a}, 0)\\}.
$$ $$
These are the only points with \\(xy = 0\\); the points with \\( y These are the only points with \\(xy = 0\\); the points with \\( y
\neq 0 \\) are \\(2\\)-torsion. The \\(\mathcal \neq 0 \\) are \\(2\\)-torsion. The \\(\mathcal
E\_{a,d}[4]\\)-coset of \\(P = (x,y)\\) is then E\_{a,d}[4]\\)-coset of \\(P = (x,y)\\) is then
$$ $$
P + \mathcal E\_{a,d}[4] = \\{ (x,y),\; (y/\sqrt a, -x\sqrt a),\; (-x, -y),\; (-y/\sqrt a, x\sqrt a)\\}. P + \mathcal E\_{a,d}[4] = \\{ (x,y),\\; (y/\sqrt a, -x\sqrt a),\\; (-x, -y),\\; (-y/\sqrt a, x\sqrt a)\\}.
$$ $$
Notice that if \\(xy \neq 0 \\), then exactly two of Notice that if \\(xy \neq 0 \\), then exactly two of
these points have \\( xy \\) non-negative, and they differ by the these points have \\( xy \\) non-negative, and they differ by the
\\(2\\)-torsion point \\( (0,-1) \\). This means that we can select \\(2\\)-torsion point \\( (0,-1) \\). This means that we can select
a representative modulo \\(\mathcal a representative modulo \\(\mathcal E\_{a,d}[2] \\)
E\_{a,d}[2] \\) by requiring \\(xy\\) nonnegative and \\(y \neq by requiring \\(xy\\) nonnegative and \\(y \neq
0\\), and we can ensure this condition by conditionally adding a 0\\), and we can ensure this condition by conditionally adding a
\\(4\\)-torsion point if \\(xy\\) is negative or \\(y = 0\\). \\(4\\)-torsion point if \\(xy\\) is negative or \\(y = 0\\).