Research proof index

A word u is within the uniqueDecodingRadius of a code C if and only if there exists exactly one codeword v in C that is that close.

A word u is close to a code C within the absolute unique decoding radius if and only if it is close within the relative unique decoding radius.

Definition 4.11.2 Combine(d*, r, (f_0, d_0), …, (f_{m-1}, d_{m-1}))(x) := if (r * φ(x)) = 1 then sum_{i < m} r_i * f_i(x) * (dstar - degree + 1) else sum_{i < m} r_i * f_i(x) * (1 - r * φ(x)^(dstar - degree + 1)) / (1 - r * φ(x))

Lemma 4.13 Let dstar be the target degree, f₁,...,f_{m-1} : ι → F, 0 < degs₁,...,degs_{m-1} < dstar be degrees and δ ∈ (0, min{(1-BStar(ρ)), (1-ρ-1/|ι|)}) be a distance parameter, then Pr_{r ← F} [δᵣ(Combine(dstar,r,(f₁,degs₁),...,(fₘ,degsₘ)))] > err' (dstar, ρ, δ, m * (dstar + 1) - ∑ i degsᵢ)

Construct a canonical polynomial from a coefficient function Fin n → R. The coefficients are stored in an array (index i gives the coefficient of X^i) and then trimmed to remove trailing zeros. -/ def ofFn [Zero R] [BEq R] [LawfulBEq R] {n : ℕ} (f : Fin n → R) : CPolynomial R := ⟨(Raw.mk (Array.ofFn f)).trim, Raw.Trim.isCanonical_trim _⟩ section Div...

For ℓ = 1, coordinate-wise special soundness is ordinary k-special soundness: the challenge values are distinct, and there is a central vector together with k - 1 siblings differing in the single coordinate , i.e. k pairwise-distinct challenge values.