Modular Square One-Way Function: Analyzing the Algorithm for Randomness, Regularity Schematic (Codec System) and Vector Normalization

Actually, The proposed paper was built as one part working through analyzing one-way modular square which is a special case of modular exponentiation with exponent of 2, counter form of quadratic residue, for encryption and PKI implementation. Later on, Due to the amount of data regarding this analysis, it seems be better to separate it to three parts. This paper (Part-1) aims to analyze a modular square one-way function using integer factorization (IF) or approximation methods. It is supposed that such function has random characteristics. This analysis considers a new innovative idea, original as claimed, focusing in notable regularities that could be used as a trapdoor for practical applications. Such as random generator, new codec system and 3D vector normalization. At the end, different types of quadratic residue algorithms and square root will be considered.