Rot256 : Cryptography & Other Random Bits.

Basics

Modulo and evaluations

Statement:

\[ \forall z \in \mathbb{F} : f(X) \equiv f(z) \mod (X - z) \]

Proof

Rewrite \( f(X) \) as \( f(X) = c + g(X) (X - z) \). Where \( c \) is a constant term.
Note \( f(X) \equiv c \mod (X - z) \), lastly observe:

\[ f(z) = c + g(X) (X - z) = c + g(X) \cdot 0 = c \]

A

Statement:

\[ \forall z \in S : f(x) = g(x) \implies f(X) \equiv g(X) \mod \mathcal{Z}_S(X) \]

Rewrite:

\[ f(X) = h(X) + \mathcal{Z}_S(X) \cdot F(X) \]

\[ g(X) = h(X) + \mathcal{Z}_S(X) \cdot G(X) \]

Where \( \forall x \in S: h(z) = f(x) = g(x) \).

Schwartz-Zippel Lemma