Ring-Switching

Thus far, when discussing commitment, we have imposed throughout the restriction whereby everything be defined over the huge field $\mathcal{T}_\tau$. This might seem like a problem—and rightly so—since many of the polynomials we'll be interested in will be defined over small tower subfields $\mathcal{T}_\iota \subset \mathcal{T}_\tau$.

In this subsection, we describe a key cryptographic technique—introduced in the academic work [DP24]—called ring-switching. Ring-switching allows us to "materialize" small-field polynomials out of large-field ones—essentially using a virtual polynomial construction analogous to those we've already seen. This allows small-field polynomials to be treated. Indeed, small-field and large-field polynomials are related to each other by reciprocal processes we call packing and refinement. Ring-switching spins up an evaluation procedure not for a committed polynomial, but for its refinement. If we're interested in a small-field polynomial to begin with, therefore, it's safe to commit to its packing instead. After all, we can then later use ring-switching to spin up that thing's refinement—i.e., nothing other than original small-field polynomial we started with.

Binius, in holistic terms, is highly motivated by the problem of efficiently treating small-field polynomials. In this section, therefore, we enter upon some of the most essential mathematical content of this entire site. This content, we note, is fairly difficult—but not unnecessarily so. Indeed, we've put in extensive effort to distill our techniques as much as possible—so as to obtain a minimized, highly efficient scheme.

Subsection Directory

This subsection contains the following pages:

• The Refinement Polynomial. Here, we introduce the relatively straightforward mathematical notions of refinement and packing. These are essential even to understand what ring-switching does (leave aside how it does it).
• The Tensor Algebra. Here, we drop a crash-course on a mathematical object called the tensor algebra, introduced in [DP24]. This algebraic object manages to exactly capture the "bookkeeping" needed to make ring-switching work.
• Motivational Remarks. Here, we set the stage with a few motivating remarks and some discussion. This page is designed to build some key intuitions.
• The Procedure. At this point, we have enough preliminaries to describe the actual algorithm of ring-switching.