Authors :
Ahmed Mohammed Al-Fahdi
Volume/Issue :
Volume 10 - 2025, Issue 1 - January
Google Scholar :
https://tinyurl.com/ytvv5h2z
Scribd :
https://tinyurl.com/2xwbpraa
DOI :
https://doi.org/10.5281/zenodo.14908867
Abstract :
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.
Keywords :
Square Root Algorithm, Modular Square, Modular Arithmetic, Quadratic Nonresidue Square Root, One-Way Function, Newton-Raphson Method, Exponentiation By Squaring, Jacobi Prime, Legendre Symbol, Fast Inverse Square Root (Invsqrt), Vector Normalization.
References :
- Prof . Dr. Rolf Oppliger,Contemporary to Cryptography,Infogaurd,Bar Switzerland ,2013
- W. Diffie and M. Hellman”New Direction in Cryptography”,IEEE Trans, 1976
- Center for education in mathematics and computing, Grade 6 Math Circles -Modular Arithmetic,University of Waterloo,2020
- Catalano,Fiore,Rosario,Vamvourellis,Algebraic (Trapdoor) One-Way Functions and their Applications, University of Catania
- Evan Huynh,Rabin’s Cryptosystem, Department of Mathematics, Linnaeus University, Sweden 2021
- Anca-Maria Nica ,Quadratic Residues and Applications in Cryptography, Alexandru Ioan Cuza Universithy of Iasi, 2020
- Pranav Gokhale ,Implementation of Square Root Function Using Quantum Circuits, Princton University Class, 2015.
- F.B. Roodenburg,Riemann’s Explicit Formula and the Prime Number Theorem, Delft Institute of Applied Mathmatics, 2020
- Dushyant,Patrick,Nikolay, Non Intrusive Codec Identification Algorithm ,Marsh 2012
- Andrei Seymour-Howell,Fast inverse square-root program,2021
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.
Keywords :
Square Root Algorithm, Modular Square, Modular Arithmetic, Quadratic Nonresidue Square Root, One-Way Function, Newton-Raphson Method, Exponentiation By Squaring, Jacobi Prime, Legendre Symbol, Fast Inverse Square Root (Invsqrt), Vector Normalization.
