🤖 AI Summary
This work addresses the challenge of efficiently encoding and verifying mathematical proofs in zero-knowledge settings. We propose a structured representation based on $2 imes2$ matrix-valued polynomials: formal statements are mapped to multivariate polynomial matrices with integer coefficients, and compact numeric fingerprints are generated via random evaluation over a finite field; matrix homomorphism enables algebraic execution of logical operations—including hypothetical reasoning and variable substitution. Our key contribution is the first encoding of formula syntactic structure into computable polynomial-matrix “fingerprints”, supporting incremental, zero-knowledge proof derivation and verification. Experiments demonstrate that only the fingerprints of initial axioms require explicit verification; all subsequent formulas are derived efficiently from ancestral fingerprints, drastically reducing both communication and computational overhead in verification.
📝 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.