# 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; $R(x) = rand\_polynomial * vanishing\_polynomial$
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 $S = \{α_1, α_2, ..., α_n\}$ is:

$$ Z(X)=∏_{i=1}^n(X−α_i) $$

impling these properties;
1. $Z(α_i)=0$ for all $α_i \in S$
2. $Z(X)≠ 0$ 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.