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Henry de Valence 2018-04-04 11:29:36 -07:00
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docs/ristretto-notes.md
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@ -1,7 +1,7 @@
Below are some notes on Ristretto, which are not an authoritative Below are some notes on Ristretto, which are not an authoritative
writeup and which may have errors. See also the [Decaf writeup and which may have errors. See also the [Decaf
paper][decaf_paper], the [libdecaf paper][decaf_paper], the [libdecaf
implementation][ristretto_libdecaf], and the [Sage implementation of Ristretto][ristretto_libdecaf], and its [Sage
script][ristretto_sage]. script][ristretto_sage].
Decaf constructs a prime-order group from a cofactor-\\(4\\) Edwards Decaf constructs a prime-order group from a cofactor-\\(4\\) Edwards
@ -22,19 +22,17 @@ see the [Decaf paper][decaf_paper] or
[_Jacobi Quartic Curves Revisited_][hwcd_jacobi] by Hisil, Wong, [_Jacobi Quartic Curves Revisited_][hwcd_jacobi] by Hisil, Wong,
Carter, and Dawson). Carter, and Dawson).
When \\(e = a\^2\\), \\(\mathcal J\_{e,A}\\) has full When \\(e = a\^2\\) is a square, \\(\mathcal J\_{e,A}\\) has full
\\(2\\)-torsion (i.e., \\(\mathcal J[2] \cong \mathbb Z /2 \times \\(2\\)-torsion (i.e., \\(\mathcal J[2] \cong \mathbb Z /2 \times
\mathbb Z/2\\)), and \mathbb Z/2\\)), and
we can write the \\(\mathcal J[2]\\)-coset of a point \\(P = we can write the \\(\mathcal J[2]\\)-coset of a point \\(P =
(s,t)\\) as (s,t)\\) as
$$ $$
P + \mathcal J[2] = \left\\{ P + \mathcal J[2] = \left\\{
(s,t), (s,t),
(-s,-t), (-s,-t),
(1/as, -t/as\^2), (1/as, -t/as\^2),
(-1/as, t/as\^2) (-1/as, t/as\^2)
\right\\}. \right\\}.
$$ $$
Notice that replacing \\(a\\) by \\(-a\\) just swaps the last two Notice that replacing \\(a\\) by \\(-a\\) just swaps the last two