Polynomial Fingerprinting for Trees and Formulas

TL;DR

This paper introduces a polynomial fingerprinting method for trees and formulas that simplifies proof steps in zero-knowledge proofs by transforming formal sentences into matrix representations and evaluating them over finite fields.

Contribution

It presents a novel approach to encode formulas as matrices and evaluate them over finite fields, enabling efficient zero-knowledge proof techniques.

Findings

Transform formal sentences into matrix representations.

Enable easy computation of proof steps via homomorphic properties.

Facilitate application of zero-knowledge methods on numeric sequences.

Abstract

To cater to the needs of (Zero Knowledge) proofs for (mathematical) proofs, we describe a method to transform formal sentences in 2x2-matrices over multivariate polynomials with integer coefficients, such that usual proof-steps like modus-ponens or the substitution are easy to compute from the matrices corresponding to the terms or formulas used as arguments. By evaluating the polynomial variables in random elements of a suitably chosen finite field, the proof is replaced by a numeric sequence. Only the values corresponding to the axioms have to be computed from scratch. The values corresponding to derived formulas are computed from the values corresponding to their ancestors by applying the homomorphic properties. On such sequences, various Zero Knowledge methods can be applied.