Thoughts on folding schemes
by Danny Willems
Morning thoughts while talking to some friends.
The “relaxation” process in folding schemes are mostly about transforming polynomials into an homogeneous form of a certain degree, see o1Labs documentation. For instance, Nova introduced the concept of relaxation for degree 2 polynomials.
On the other side, jacobian coordinates and projective coordinates of elliptic
curves (as a reminder, elliptic curves points are zeroes of multivariate
polynomials - for instance P(X, Y) = X^3 + a X + b - Y^2) are homogeneous forms
of the polynomial equations describing the curve. For instance, projective
coordinates transform P(X, Y) = X^3 + a X + b - Y^2 into a polynomial
P(X, Y, Z) = X^3 + a Z^2 X + b Z^3 - Z Y^2.
When we are making an IVC proof, we are aggregating randomized commitments, which are elliptic curves points of a prime order. Therefore, there are of the form g^x, with x being a scalar field, and g^x is a value that can be represented as a triplet satisfying an homogeneous polynomial. Thought stops here, but I’m wondering if we can make a link between both. And also if there are some optimisations possible for the IVC verifier.
Folding schemes allow to only focus on making a proof of the computation of
P(X' + r X, Y' + r Y, Z' + r Z) which englobes the computation of P(r X, r Y,
r Z) and P(X', Y', Z'), plus an additional error term we “ignore”, defined by
a polynomial E(X, Y, Z, r X', r Y', r Z').
When we take rX, rY, rZ, if Z is the “u” in Nova, we take a certain point (a, b, c) from the class representing the value (x, y).
Links:
- Higher geometry; an introduction to advanced methods in analytic geometry
- [Duality (projective geometry)](https://en.wikipedia.org/wiki/Duality_(projective_geometry)