The Inception of Polynomial Interactive Oracle Proof (IOP) and its Implementation

Lecture 4.5

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Structure:

Introduction

Understanding the Polynomial IOP

The Two-Step Plan

Implementation of the Two-Step Plan

The Polynomial IOP for Circuit Satisfiability

Conclusion

Introduction:

As advancements in computational complexity theory continue to unravel, the role of interactive proofs and polynomial computations take center stage in the development of efficient and secure protocols. At the heart of this revolution lies the Polynomial Interactive Oracle Proof (IOP), a mechanism designed to allow the verifier to confirm the authenticity of a prover's transcript in a time-efficient and resource-conserving manner. This article delves into the underlying principles of the polynomial IOP, shedding light on how the protocol establishes a trustworthy relationship between the prover and the verifier, and highlighting the pivotal role of extension polynomials in this process. In the grand scheme of computer science, these principles establish a pathway for more robust and secure verification protocols.

The Start of the Polynomial IOP

The Polynomial Interactive Oracle Proof (IOP) essentially uses a protocol based on the sum check protocol. It allows the verifier to confirm that the prover possesses a correct transcript, which we interpret as a function mapping from the domain 0,1 to the log_s. In this context, a 'transcript' is a record of computation, where one or more messages from the prover specify a large polynomial that the verifier will evaluate at a single point, but will not read in full.

The prover's first message in the polynomial IOP claims to extend a correct transcript. The extension polynomial, h, is supposed to extend the correct transcript such that if you feed into h a Boolean input (an input in 0,1 to the log_s), it produces the same output as the transcript at that input. The verifier's challenge is to confirm that this polynomial, h, sent by the prover indeed extends the correct transcript.

However, due to the constraints of the polynomial IOP, the verifier can only evaluate the polynomial, h, at a limited number of evaluation points, while verifying that h extends the correct transcript involves s many evaluations of h. Therefore, the design of a clever protocol is necessary to allow the verifier to confirm that h(x) equals T(x) for all x in 0,1 of the log_s for a correct transcript T.

Intuition for why h is a useful object for P to send

The polynomial extension h of a correct transcript is a useful object for the prover to send due to its distance amplifying nature. It effectively serves as an amplified encoding of the correct transcript, T. Even if two transcripts disagree at a single input, their extension polynomials will disagree at almost all inputs in the field to the log_s. Consequently, even tiny discrepancies in transcripts can be easily detected by a verifier who is only allowed to evaluate these extension polynomials at a single point or a few points.

Reminder: The Start of the Polynomial IOP

To remind you, the prover's first message in the polynomial IOP is a polynomial h, claimed to extend a correct transcript T. The verifier needs to confirm that h indeed extends some correct transcript but is only able to learn a few evaluations of h.

Two-Step Plan of Attack

In order to tackle this problem, we adopt a two-step plan. The first step is to transform h into a related polynomial, g_sub h. This related polynomial has three times as many variables as h and is required to have two properties:

Each extending a correct transcript should be equivalent to g_sub h vanishing over the Boolean hypercube. This means that if you input into g_sub h any Boolean value (any triple a,b,c with each of a,b,c being a gate label), g_sub h should output zero.

Evaluating g_sub h at any input r should only necessitate the evaluation of h itself at three inputs. This ensures that the verifier has manageable 'query access' to g_sub h.

This transformation simplifies the problem considerably as the verifier can now just check that g_sub h equals zero at every single Boolean input.

The second step in our plan of attack is to design an interactive proof that indeed checks that g_sub h vanishes over the Boolean hypercube. This interactive proof should only require the verifier to evaluate g_sub h at a random point r.

Step 1 of the plan

In this step, the objective is to work with a given polynomial 'h' and derive another polynomial 'g' subscript h (g_h) in thrice the number of variables such that the given polynomial 'h' extends a correct transcript if and only if 'g_h' vanishes over the Boolean hypercube. This step introduces two functions, add_twiddle(a,b,c) and mult_twiddle(a,b,c), which are multilinear extensions of wiring predicates in a given circuit.

add_twiddle(a,b,c): This function takes three Boolean gate labels, a, b, and c. It returns 1 if 'a' is an addition gate whose inputs are gates labeled 'b' and 'c'. Otherwise, it returns 0.

mult_twiddle(a,b,c): Similar to add_twiddle, this function returns 1 if 'a' is a multiplication gate whose inputs are gates labeled 'b' and 'c'. Otherwise, it returns 0.

These functions help in defining 'g_h' in such a way that it evaluates to zero for all triples of inputs (a,b,c) if and only if the values assigned to each gate in the circuit respect the operations of that gate.

Step 2: A Hint

In this step, the focus is on verifying that the polynomial 'g_h' vanishes over the Boolean hypercube. A concept called the Vanishing Polynomial is introduced, which is the lowest degree polynomial that evaluates to zero over all points in a subset of a finite field. If 'g_h' is divisible by the vanishing polynomial for a set 'H', then 'g_h' vanishes over the set 'H'. However, the issue is that 'g_h' is not univariate but has multiple variables, and working over the integers would make the integers too large, increasing the proof size.

The actual protocol

The actual protocol uses an interactive proof called the Sum-check protocol to confirm that 'g_h' vanishes over the Boolean hypercube. The Sum-check protocol allows the verifier to force the prover to sum up all evaluations of a multivariate polynomial over all inputs, with a proof length proportional to the total degree of the polynomial.

The polynomial 'g_h' is put through the Sum-check protocol to ensure it vanishes over the Boolean hypercube. The protocol can handle 'g_h' having multiple variables and has efficiency benefits, such as the prover not needing to send large quotient polynomials.

Sum-check protocol: a reminder

The Sum-check protocol is briefly recalled. It is designed to verify sums of polynomial evaluations. The protocol proceeds in rounds, where the prover sends messages to the verifier. The verifier's time is often dominated by the time required to evaluate the polynomial at a single randomly chosen point.

The polynomial IOP for circuit satisfiability

To ensure that 'g_h' vanishes at all inputs in the Boolean hypercube, the Sum-check protocol is applied to compute the sum of squared evaluations of 'g_h' over the Boolean hypercube. The sum should be zero if 'g_h' correctly vanishes.

This is done interactively where the verifier needs to evaluate 'g_h' at a single random point at the end of the Sum-check protocol. The verifier runs in time logarithmic in circuit size, and the prover can compute all the messages in the Sum-check protocol in linear time relative to the size of the circuit.

In summary, this plan combines polynomial manipulation with an interactive proof protocol to verify that the polynomial 'g_h' accurately represents the operations of a circuit and vanishes over the Boolean hypercube, which is necessary to extend a correct transcript.

Conclusion:

To summarize, the Polynomial Interactive Oracle Proof represents a significant step forward in the arena of computational complexity, leveraging the power of polynomial computations and interactive proofs to streamline and bolster the verification process. In its core, the protocol not only ingeniously capitalizes on the properties of extension polynomials to validate the prover's transcript but also employs the Sum-check protocol to assure the accuracy of multivariate polynomials. The Polynomial IOP, with its efficient two-step plan, empowers the verifier to perform their task in logarithmic time relative to the circuit size, underscoring the potential for scalability in larger computational systems. As we move towards an era of even greater computational needs, the principles and strategies outlined in the Polynomial IOP pave the way for future breakthroughs in efficient and secure verification protocols.