One actual affine breaker for every fourth-order parity test

With C candidates per coordinate, every test uses the same selected affine breaker with 4 * C - 1 tampered slots. Its seed and output widths therefore do not depend on the tested coordinate subset. The Boolean component is the first bit of this fixed full-output construction.

This is the fixed-family indexing in Chattopadhyay--Liao, Extractors for Sum of Two Sources, Lemma 5.4, equation (5): https://arxiv.org/abs/2110.12652.

def Algebraic.Cutwidth.Extractor.affineTestSeedBits (n C a target : ℕ) : ℕ

The common seed width for all coordinate tests with C candidates.

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def Algebraic.Cutwidth.Extractor.affineTestOutputBits (n C a target : ℕ) : ℕ

The common full-output width for the fixed test breaker.

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def Algebraic.Cutwidth.Extractor.affineTestBreaker (n C a target : ℕ) (x : Fin n → Bool) (y : Fin (affineTestSeedBits n C a target) → Bool) (advice : List Bool) : Fin (affineTestOutputBits n C a target) → Bool

The same actual selected affine breaker is used for every test and candidate.

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def Algebraic.Cutwidth.Extractor.affineTestFirst (n C a target : ℕ) : Fin (affineTestOutputBits n C a target)

The canonical first output coordinate of the fixed test breaker.

Equations

• Algebraic.Cutwidth.Extractor.affineTestFirst n C a target = ⟨0, ⋯⟩

def Algebraic.Cutwidth.Extractor.affineTestBit (n C a target : ℕ) (x : Fin n → Bool) (y : Fin (affineTestSeedBits n C a target) → Bool) (advice : List Bool) : Bool

The actual Boolean component used by the XOR source reduction.

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