What is Algebraic Intermediate Representation (AIR)

July 26th, 2022

**Fox is an Ethereum zkRollup using ZK-EVM and its original ZK-FOAKS technology. ZK-STARKS comes from Zero Knowledge - Fast Objective Argument of Knowledges that is faster than Stark with linear proof time, sublinear verification time, transparent, and proof recursion inside.**

Arithmetization is the simplification of computed statements to a set of algebraic statements involving a set of bounded degree polynomials. In STARK Protocol, the arithmetical output is an algebraic intermediate representation of a computation. A formal definition of AIR includes the following three elements:

(1) Execution trace is a two-dimensional matrix, where each line represents the state of a calculation at time points, each column corresponds to an algebraic register tracked over all the steps of the calculation. Let 𝑇 denote execution trace matrix.

(2) Transition constraints that define algebraic relations between two (or more) rows of the execution trace.

(3) Boundary constraints that enforce equality between some cells of execution and a set of constant values. Boundary constraints can be thought of as defining a set of constraints calculated input and output values.

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