Mathematical Hash Functions
Algorithms that produce a fixed-size output (hash) from an input of any size, used for data integrity verification and password storage.
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The statement of the theorem
Define a hash function $H: \Sigma^* \to \\{0, 1\}^n$ (where the output length is fixed). The function must satisfy three properties: \n1. \textbf{Pre-image Resistance:} Given $h \in \text{Range}(H)$, finding $M$ such that $H(M) = h$ must be computationally infeasible.\n2. \textbf{Second Pre-image Resistance:} Given $M_1$, finding $M_2 \neq M_1$ such that $H(M_2) = H(M_1)$ must be computationally infeasible.\n3. \textbf{Collision Resistance:} Finding any pair $(M_1, M_2)$ such that $M_1 \neq M_2$ and $H(M_1) = H(M_2)$ must be computationally infeasible.
Source: Wikipedia