Chapter3 - Definitions and Technical Preliminaries¶

3.1 Interactive Proofs¶

In an interactive proof, the Prover (P) claims having a value \(y = f(x)\). To validate this claim, the Verifier (V *) exchanges a series of messages with *P. At the conclusion of this protocol, V must output either 0 or 1, where 1 indicates acceptance of the prover’s claim (\(y = f(x)\)) and 0 indicates rejection. The entire sequence of k messages \(t := (m_1, m_2, ..., m_k)\) exchanged by P and V, along with the claimed answer y, is termed a transcript.

The output of verifier \(V\) on input \(x\) during interaction with a deterministic prover strategy P, with V’s internal randomness equal to \(r\), is denoted as \(Out(V,x,r,P) \in \{0,1\}\).

An interactive proof system (V,P) is considered to have completeness error \(δ_c\) and soundness error \(δ_s\):

Completeness error \(δ_c\): \(\text{Pr}[\text{out}(V, x, r, P) = 1]\) \(\ge\) 1 - \(δ_c\)
Soundness error \(δ_s\): \(\text{Pr}[\text{out}(V, x, r, P') = 1]\) \(\le\) \(δ_s\), where \(P'\) is a prover strategy with value \(y \ne f(x)\) (statistical soundness).

An interactive proof system is deemed valid if \(δ_c\) and \(δ_s\) are both ≤ ⅓.

3.2 Argument Systems¶

Definition: Read here.

3.3 Robustness of Definitions and the Power of Interaction¶

Any IP for a function \(f\) with \(δ_c\) ≤ ⅓ can be transformed into perfect completeness (\(δ_c\) = 0), with a polynomial blowup in the verifier’s costs.
Soundness Error: \(1 / |F|\), where \(F\) is the field over which the interactive proof is defined.

Interactive Proofs for Languages Versus Functions¶

The standard requirements of an IP for the language \(L\) are:

Completeness: For any \(x \in L\), some prover strategies will cause the verifier to accept with high probability.
Soundness: For any \(x \notin L\), then for every prover strategy, the verifier rejects with high probability.

These properties do not necessitate convincing refutations (convincing proofs of falsity) for false statements (inputs not in the language).

NP and IP¶

Refer to the definitions here: IP and NP.

3.4 Schwartz-Zippel Lemma¶

Read the lemma here.

3.5 Low Degree and Multilinear Extensions¶

Read the definitions of Low-degree extension, Multilinear and Extension.

Fact 3.5: Any function \(f: \{0, 1\}^v → F\) has a unique multilinear extension (MLE) over \(F\). The proof is available here.

We use the notation \(\tilde{f}\) for this special extension of \(f\):

\[ \begin{aligned} \tilde{f}(x_1, ..., x_v) = \sum_{w\in \{0,1\}^v} f(w)· χ_w(x_1, ..., x_v) \end{aligned} \]

where, for any \(w = (w_1,...w_v)\):

\[ \begin{aligned} χ_w(x_1,...,x_v) := \prod_{i=1}^v (x_iw_i + (1 - x_i)(1-w_i)) \end{aligned} \]

The set \(\{χ_w: w \in \{0, 1\}^v\}\) is referred to as the set of multilinear Lagrange basis polynomials.