# do255e / jq255e

The curve do255e is a GLV curve, i.e. with equation $$y^2 = x(x^2 + b)$$. Its Jacobi quartic counterpart, jq255e, has equation $$e^2 = -4bu^4 + 1$$. We apply the following criteria:

• Modulus $$q = 2^{255} - m$$ should be equal to 5 modulo 8. For the GLV curve not to be supersingular, we need $$q$$ to be equal to 1 or 5 modulo 8; the second choice makes computation of square roots in the field easier to implement.

• Since the $$j$$-invariant of such a curve is fixed (it's 1728), there are only two potential curves (up to isomorphisms) to check for a given field. We can thus always enforce that $$b = 2$$ or $$-2$$, which are convenient values for implementation.

• Curve order must be equal to $$2r$$ for a prime integer $$r$$.

Under these criteria, the first match is for $$m = 18651$$. Here are the resulting curve parameters:

• Name: do255e / jq255e
• Field: integers modulo $$q = 2^{255} - 18651$$
• Equations: $$y^2 = x(x^2 - 2)$$ and $$e^2 = 8u^4 + 1$$
• Order: $$2r$$, with $$r = 2^{254} - 131528281291764213006042413802501683931$$
• Generator: $\begin{eqnarray*} G_x &=& 2 \\ G_y &=& 2 \\ G_e &=& 3 \\ G_u &=& 1 \end{eqnarray*}$