Pallas, Vesta, and Zcash

Yesterday's post mentioned Tweedledum and Tweedledee, a pair of elliptic curves named after characters in Lewis Carroll's Through the Looking Glass.

Zcash uses a very similar pair of elliptic curves for zero-knowledge proofs: Pallas and Vesta. One named after a Greek goddess associated with war, and one named after a Roman goddess associated with home and hearth. (Zcash used the Jubjub curve, also mentioned yesterday, though not in the most recent release.)

The curves Pallas and Vesta are collectively known as the pasta" curves, pasta being a portmanteau ofpallas andvesta.

pa(llas ve)sta

Tweedledum, Tweedledee, Pallas, and Vesta all have the equation

y^2 =x^3 + 5

over different prime-order fields.

The number of elements in Tweedledum's field equals the number of elements in Tweedledee's elliptic curve, and vice versa. The same relationship holds between Pallas and Vesta. And all four curves have roughly 2²⁵⁴ points.

If p is the size of Tweedledum's field (and Tweedledee's curve), and q is the size of Tweedledee's field (and Tweedledum's curve),

p= 2²⁵⁴+ 4707489545178046908921067385359695873
q= 2²⁵⁴+ 4707489544292117082687961190295928833

If p is the size of Pallas' field (and Vesta's curve), and q is the size of Vesta's field (and Pallas' curve),

p= 2²⁵⁴ + 45560315531419706090280762371685220353
q= 2²⁵⁴ + 45560315531506369815346746415080538113

The post Pallas, Vesta, and Zcash first appeared on John D. Cook.