Concept Of Blinding Polynomials in ZKP

Blinding polynomials are mathematical constructs used in zero-knowledge proof systems, particularly in polynomial commitment schemes, to hide or "blind" the actual polynomials being proved while maintaining their mathematical properties and structure. A blinding polynomial is essentially a random polynomial that's added to the original polynomial to mask its coefficients, the resulting sum preserves the mathematical properties needed for the proof while hiding the actual values.

Mathematical Representation: Let's say you have an original polynomial

$P(x)$. The blinded version would be:

$P_{blinded}(x)=P(x)+R(x)$ where

$R(x)$ is the random blinding polynomial.

The blinding polynomial

$R(x)$ needs to be;

1. Choosen randomly
2. Have the same of higher degree as
   $P(x)$ to properly mask all coefficients
3. The Coeffiecients of
   $R(x)$ are choosen from a large field to ensure security

$R(x)$ needs to be constructed in such a way that on adding it to

$P(x)$ the structure of

$P(x)$ remains; this can be achieved by;

1. generating the randoms field elements using it as the coeffiecents to create a polynomial rand_polynomial.
2. Multiply this polynomial with the vanishing_polynomial; it is important to note, vanishing polynomial construction varies with the nature of domain of the polynomial (e.g. Real numbers, Roots of unity and so on).
3. The blinding_polynomial is not given by;
   $R(x)=rand\mathrm{\_}polynomial\ast vanishing\mathrm{\_}polynomial$
4. To obtain the blinded_polynomial, perform this operation;
   $P_{blinded}(x)=P(x)+R(x)$.

Constructing Vanishing Polynomial when Domain = Real Numbers The vanishing polynomial

$Z(X)$ for a set of points

$S=\{{\alpha }_1,{\alpha }_2,...,{\alpha }_n\}$ is:

$$Z(X)={\prod }_{i=1}^n(X-{\alpha }_i)$$

impling these properties;

1. $Z({\alpha }_i)=0$ for all
   ${\alpha }_i\in S$
2. $Z(X)\ne 0$ for all
   $X\notin S$
3. Degree of
   $Z(X)$ is
   $|S$| (size of the set)

Constructing Vanishing Polynomial when Domain = Roots of Unity Constructing the vanishing polynomial in this domian is alot more less complicated, it can be represented by the given equation;

$$Z_H(X)=X_n-1$$ Where

$n$ is the size of the evaluation domain (expressed as roots of unity).

Understanding blinding polynomials is crucial for implementing secure and private zero-knowledge proof systems while maintaining the mathematical integrity of the underlying computations.