diff --git a/docs/ristretto-notes.md b/docs/ristretto-notes.md index b558b74..f88979e 100644 --- a/docs/ristretto-notes.md +++ b/docs/ristretto-notes.md @@ -52,20 +52,23 @@ 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_][hwcd_edwards] of Hisil, Wong, Carter, -and Dawson. These correspond to the affine model - -$$\mathcal E\_{a,d} : ax\^2 + y\^2 = 1 + dx\^2y\^2.$$ - +The primary internal model in `curve25519-dalek` for Curve25519 points +is the [_Extended Twisted Edwards Coordinates_][hwcd_edwards] of +Hisil, Wong, Carter, and Dawson. These correspond to the affine model +$$ +\mathcal E\_{a,d} : ax\^2 + y\^2 = 1 + dx\^2y\^2. +$$ In projective coordinates, we represent a point as \\((X:Y:Z:T)\\) -with $$XY = ZT, \quad aX\^2 + Y\^2 = Z\^2 + dT\^2.$$ (For more -details on this model, see the documentation for the `edwards` -module). The case \\(a = 1\\) is the _untwisted_ case; we only -consider \\(a = \pm 1\\), and in particular we focus on the twisted -Edwards form of Curve25519, which has \\(a = -1, d = --121665/121666\\). When not otherwise specified, we write -\\(\mathcal E\\) for \\(\mathcal E\_{-1, -121665/121666}\\). +with +$$ +XY = ZT, \quad aX\^2 + Y\^2 = Z\^2 + dT\^2. +$$ +(For more details on this model, see the +[`curve_models`][curve_models] documentation). The case \\(a = 1\\) is +the _untwisted_ case; we only consider \\(a = \pm 1\\), and in +particular we focus on the twisted Edwards form of Curve25519, which +has \\(a = -1, d = -121665/121666\\). When not otherwise specified, +we write \\(\mathcal E\\) for \\(\mathcal E\_{-1, -121665/121666}\\). When both \\(d\\) and \\(ad\\) are nonsquare (which forces \\(a\\) to be square), the curve is *complete*. In this case the @@ -333,4 +336,5 @@ isogeny \\(\hat{\theta}\\). Defer this for now. [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 \ No newline at end of file +[twisted_edwards]: https://eprint.iacr.org/2008/013.pdf +[curve_models]: ../../curve_models/index.html \ No newline at end of file