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docs/ristretto-notes.md
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@ -1,15 +1,25 @@
Below are some notes on Ristretto, which are *NOT* a full writeup and which may have errors.
Below are some notes on Ristretto, which are not an authoritative
writeup and which may have errors. See also the [Decaf
paper][decaf_paper], the [libdecaf
implementation][ristretto_libdecaf], and the [Sage
script][ristretto_sage].
# Notes on Ristretto
Decaf constructs a prime-order group from a cofactor-\\(4\\) Edwards
curve by defining an encoding of a related Jacobi quartic, then
transporting the encoding from the Jacobi quartic to the Edwards curve
by means of an isogeny. Ristretto uses a different Jacobi quartic and
a different isogeny, but is otherwise similar.
These notes only describe Ristretto, and focus on the cofactor-\\(8\\)
case.
## The Jacobi Quartic
The Jacobi quartic is parameterized by \\(e, A\\), and is of the
The Jacobi quartic curve is parameterized by \\(e, A\\), and is of the
form $$ \mathcal J\_{e,A} : t\^2 = es\^4 + 2As\^2 + 1, $$ with
identity point \\((0,1)\\). For more details on the Jacobi quartic,
see the [Decaf paper](https://eprint.iacr.org/2015/673.pdf) or
[_Jacobi Quartic Curves
Revisited_](https://eprint.iacr.org/2009/312.pdf) by Hisil, Wong,
see the [Decaf paper][decaf_paper] or
[_Jacobi Quartic Curves Revisited_][hwcd_jacobi] by Hisil, Wong,
Carter, and Dawson).
When \\(e = a\^2\\), \\(\mathcal J\_{e,A}\\) has full
@ -19,10 +29,13 @@ we can write the \\(\mathcal J[2]\\)-coset of a point \\(P =
(s,t)\\) as
$$
P + \mathcal J[2] = \left\\{
(s,t),
(-s,-t),
(1/as, -t/as\^2),
(-1/as, t/as\^2) \right\\}.
(s,t),
(-s,-t),
(1/as, -t/as\^2),
(-1/as, t/as\^2)
\right\\}.
$$
Notice that replacing \\(a\\) by \\(-a\\) just swaps the last two
points, so this set does not depend on the choice of \\(a\\). In
@ -42,9 +55,8 @@ The encoding is then the (canonical byte encoding of the)
## The Edwards Curve
Our primary internal model for Curve25519 points are the [_Extended
Twisted Edwards Coordinates_](https://eprint.iacr.org/2008/522.pdf)
of Hisil, Wong, Carter, and Dawson.
These correspond to the affine model
Twisted Edwards Coordinates_][hwcd_edwards] of Hisil, Wong, Carter,
and Dawson. These correspond to the affine model
$$\mathcal E\_{a,d} : ax\^2 + y\^2 = 1 + dx\^2y\^2.$$
@ -316,3 +328,11 @@ It's possible to do batch encoding of \\( [2]P \\) using the dual
isogeny \\(\hat{\theta}\\). Defer this for now.
## ???
[ristretto_sage]: https://sourceforge.net/p/ed448goldilocks/code/ci/master/tree/aux/ristretto/ristretto.sage
[ristretto_libdecaf]: https://sourceforge.net/p/ed448goldilocks/code/ci/master/tree/
[decaf_paper]: https://eprint.iacr.org/2015/673.pdf
[hwcd_jacobi]: https://eprint.iacr.org/2009/312.pdf
[hwcd_edwards]: https://eprint.iacr.org/2008/522.pdf
[edwards_edwards]: https://www.ams.org/journals/bull/2007-44-03/S0273-0979-07-01153-6/S0273-0979-07-01153-6.pdf
[twisted_edwards]: https://eprint.iacr.org/2008/013.pdf