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Split explanation of encoding formulas

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Henry de Valence 2018-04-04 17:42:10 -07:00
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docs/ristretto-notes.md
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@ -179,10 +179,10 @@ apply the encoding for \\( \[2\](\mathcal E) / \mathcal E[2] \\).
## The Ristretto Encoding ## The Ristretto Encoding
We can write the above encoding/decoding procedure concretely (in affine We can write the above encoding/decoding procedure in affine
coordinates) as follows: coordinates as follows:
### Encoding ### Encoding in Affine Coordinates
On input \\( (x,y) \in \[2\](\mathcal E)\\), a representative for a On input \\( (x,y) \in \[2\](\mathcal E)\\), a representative for a
coset in \\( \[2\](\mathcal E) / \mathcal E[4] \\): coset in \\( \[2\](\mathcal E) / \mathcal E[4] \\):
@ -203,8 +203,12 @@ If \\(\mathcal E\\) has cofactor \\(4\\), we skip the first step,
since our input already represents a coset in since our input already represents a coset in
\\( \[2\](\mathcal E) / \mathcal E[2] \\). \\( \[2\](\mathcal E) / \mathcal E[2] \\).
To see that this corresponds to the encoding procedure above, notice ### Interpreting the Encoding Procedure
that the first step lifts from \\( \mathcal E / \mathcal E[4] \\) to
How does this procedure correspond to the description involving
\\( \theta \\)?
The first step lifts from \\( \mathcal E / \mathcal E[4] \\) to
\\(\mathcal E / \mathcal E[2]\\). To understand steps 2 and 3, \\(\mathcal E / \mathcal E[2]\\). To understand steps 2 and 3,
notice that the \\(y\\)-coordinate of \\(\theta(s,t)\\) is notice that the \\(y\\)-coordinate of \\(\theta(s,t)\\) is
$$ $$
@ -232,7 +236,7 @@ sign (by choosing the positive square root). Finally, the check
that \\(y \neq -1\\) prevents division-by-zero when encoding the that \\(y \neq -1\\) prevents division-by-zero when encoding the
identity; it falls out of the optimized formulas below. identity; it falls out of the optimized formulas below.
### Decoding ### Decoding to Affine Coordinates
On input `s_bytes`, decoding proceeds as follows: On input `s_bytes`, decoding proceeds as follows: