Research proof index

The T-transform is natural in any oracle-semantics morphism that preserves both the plaintext-to-coins query capability and the distinguished lift of ProbComp.

An m×n matrix has full rank if the submatrix consisting of columns 1 through m has rank m.

Completeness theorem for Merkle trees: for any full binary tree with 2 ^ n leaves, and for any index i, the honestly-generated opening proof verifies against the honestly-built root with probability one.

A functional completeness theorem for Merkle proofs built from buildMerkleTree_with_hash.

[⊗_{i=0}^{ϑ-1}(1-r_i, r_i)] · [ - u₀ -; ...; - u_{2^ϑ-1} - ] - [⊗_{i=0}^{ϑ-2}(1-r_i, r_i)] · ([(1-r_{ϑ-1}) · U₀] + [r_{ϑ-1} · U₁])

For a multilinear t (each variable has degreeOf ≤ 1), substituting t into a univariate Q : L[X] via Polynomial.aeval yields a multivariate polynomial whose degree in each variable is bounded by Q.natDegree. Used by the structured sumcheck to bound the degree of Q(witness) in the round polynomial H = P · Q(t).