Explicit-Formulas Database


Genus-1 curves over large-characteristic fields

Doubling-oriented Doche–Icart–Kohel curves: y2=x3+a*x2+16*a*x
Tripling-oriented Doche–Icart–Kohel curves: y2=x3+3*a*(x+1)2
Edwards curves: x2+y2=c2*(1+d*x2*y2)
Hessian curves: x3+y3+1=3*d*x*y
Jacobi intersections: s2+c2=1, a*s2+d2=1
Jacobi quartics: y2=x4+2*a*x2+1
Montgomery curves: b*y2=x3+a*x2+x
Short Weierstrass curves: y2=x3+a*x+b
Twisted Edwards curves: a*x2+y2=1+d*x2*y2
Twisted Hessian curves: a*x3+y3+1=d*x*y

Ordinary genus-1 curves over binary fields

Binary Edwards curves: d1*(x+y)+d2*(x2+y2)=(x+x2)*(y+y2)
Hessian curves: x3+y3+1=3*d*x*y
Short Weierstrass curves: y2+x*y=x3+a2*x2+a6

The Explicit-Formulas Database is joint work by Daniel J. Bernstein and Tanja Lange, building on work by many authors.

We've completely revamped the EFD to include more coordinate systems, more formulas, fully automated formula verification, and fully automated operation counts. Latest news: The EFD also covers characteristic 2!

New paper on the EFD: "Analysis and optimization of elliptic-curve single-scalar multiplication", to appear in the proceedings of Fq8.

See also the zoo pictures.

The old Explicit-Formulas Database is still available if you want to look at it.