Understanding The Use of Polynomials in Interactive Proofs and SNARKs

Lecture 4.3

Structure:

I. Introduction

• Definition and importance of interactive proofs and SNARKs
• Overview of the role of polynomials in proof design
• Brief introduction of the topics to be covered (SZDL Lemma, multivariate extensions, multilinear extensions)

II. The SZDL Lemma: Key to Efficient Proof Verification

• Definition of the SZDL Lemma
• Importance and role of the SZDL Lemma in proof verification
• Practical examples and applications of the SZDL Lemma

III. The Power of Extension Polynomials: From Single-Variable to Multivariate

• Definition and concept of extension polynomials
• Explanation of low-degree and multivariate extensions
• Implications of these extensions in interactive proof designs
• Practical examples and applications of extension polynomials

IV. Efficient Evaluation of Multilinear Extensions: Enhancing the Feasibility of Interactive Proofs

• Explanation of the importance of evaluating multilinear extensions
• Techniques to perform these evaluations efficiently
• Impact of efficient evaluations on the overall performance of proof verification
• Practical examples and applications of efficient evaluation

V. Conclusion

• Recap of the importance of multivariate polynomials in interactive proof design
• Recap of the key concepts covered (SZDL Lemma, extension polynomials, efficient evaluation)
• Final remarks on how these concepts contribute to the creation and transformation of SNARKs.

Introduction:

In the world of interactive proofs and succinct non-interactive arguments of knowledge (SNARKs), the importance of multivariate polynomials cannot be overstated. These mathematical constructs, along with the concepts of extension polynomials, form the backbone of proof design in cryptographic commitment schemes. This article delves into the Schwarz-Zippel-De Millo-Lipton Lemma (SZDL Lemma), explains low-degree and multilinear extensions, and elaborates on how to evaluate multilinear extensions quickly. By understanding these integral concepts, one can comprehend how interactive proofs are made efficient and how they are transformed into SNARKs.

Recap: SZDL Lemma

The SZDL Lemma, also known as the Schwarz-Zippel-De Millo-Lipton Lemma, is a critical technical concept often used when designing efficient interactive proofs that are later turned into snarks. The Lemma is centered on the properties of low-degree polynomials. Simply put, if we have two distinct univariate polynomials of degree at most d, the chance that they will agree at a randomly selected input is at most their degree over the size of the field. This Lemma extends to the multivariate case, stating that for any two distinct l-variate polynomials of total degree at most d, the probability that they will agree at a randomly selected point is again at most their total degree over the size of the field.

The SZDL Lemma is vital in interactive proof design as it enables the use of multivariate polynomials, which helps keep the total degree of the polynomials within the protocol low. This allows the proofs to remain short and fast to verify. On the contrary, using univariate polynomials would result in large, time-consuming proofs. Therefore, the Lemma provides the foundation for l-variate functions, which we will delve into next.

Low-Degree and Multilinear Extensions

To build efficient interactive proofs, we often employ multivariate polynomials. Consider a function f defined over l variables, taking in l 0-1 valued inputs and mapping them to a field element. An l-variate polynomial g, defined over a field F, is said to extend f if f and g agree at any point where f is defined. This notion of extension allows us to expand the domain of f from 0-1 to the l, to the field to the l.

A fascinating fact is that any function f with domain 0-1 to the l has a unique extension that's multilinear, i.e., the degree of the polynomial is at most one in each variable. This multilinear extension of f, denoted by f twiddle, is a distance-amplified encoding of f. This means that if two functions, f and f', disagree even at one input out of all 2 to the l inputs, their multilinear extension polynomials will disagree at nearly all inputs in the field to the l. This property of amplifying small differences into significant, easily detectable ones is precisely why multilinear extension polynomials are beneficial in interactive proof design.

Evaluating Multilinear Extension Quickly

Interactive proofs often require both the prover and verifier to quickly evaluate these multilinear extensions at a number of points. Given the evaluations of a function f mapping 0-1 to the l to a field, there exists an algorithm that can evaluate the multilinear extension of f at any desired point r in the field to the l, running in time order 2 to the l.

The algorithm uses the process of LaGrange interpolation. For each input in 0-1 to the l, we can define a LaGrange basis polynomial that maps the input to one and all other inputs from 0-1 to the l to zero. The multilinear extension of f can then be expressed as a weighted sum of these LaGrange basis polynomials, giving an explicit expression for the extension at any desired point r.

This comprehensive view of the SZDL Lemma and multilinear extensions provides a robust foundation for understanding and creating interactive proofs, a vital aspect of modern cryptography.

Conclusion:

The use of multivariate polynomials in interactive proof design plays a pivotal role in maintaining short proofs that are quick to verify. The ability to extend the domain of a function and create multilinear extension polynomials paves the way for efficient detection of discrepancies between functions. The efficient evaluation of these multilinear extensions adds to the overall feasibility of the process. As such, the understanding of the SZDL Lemma and the extension of multivariate polynomials are indispensable in the design of interactive proofs and their conversion into SNARKs.

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