Comparative Study Of Group Structures In Modern Cryptosystems

Abstract

An Analysis Of Algebraic Group Structures Used In Contemporary Cryptosystems Is Presented In This Paper. The Multiplicative Group Of Integers Modulo N (used In RSA And Classical Diffie–Hellman), Cyclic Subgroups Of Finite Fields, Elliptic Curve Groups (used In ECC), Pairing-based Groups (bilinear Pairings On Elliptic Curves), And Other Algebraic Structures Closely Related To Group Theory That Support Post-quantum Schemes (e.g., Module/ring Structures In Lattice Cryptography And Isogeny-based Groups) Are All Covered. We Go Over The Mathematical Description, Cryptographic Applications, Security Presumptions, Algorithmic Complexity (for Group Operations And Principal Assaults), Efficiency And Implementation Factors, And Suggested Parameter Selections For Each Structure. The Study Ends With A Side-by-side Comparison That Highlights The Advantages, Disadvantages, And Potential Paths Forward.The Hardness Of Particular Computational Tasks Specified Over Algebraic Structures, Especially Groups, Is A Critical Component Of The Security Of Contemporary Public-key Cryptosystems. The Group Structures Underlying Modern Cryptographic Schemes, Such As Finite Cyclic Groups (used In Diffie-Hellman And DSA), Elliptic Curve Groups (ECC), And Newly Developed Post-quantum Structures Like Lattices, Isogenies, And Multivariate Polynomials, Are All Thoroughly Compared In This Paper. We Look At The Mathematical Underpinnings Of Each Group, Related Hard Problems, Trade-offs Between Security And Efficiency, Implementation Issues, And Defense Against Classical And Quantum Attacks. The Analysis Shows That The Emergence Of Quantum Computing Is Propelling A Shift Toward More Sophisticated Non-abelian And Structured Lattice-based Groups, Even If Elliptic Curve Groups Now Dominate Practical Deployments Because To Their Efficiency And Compactness. In Order To Help Choose Group Structures For Upcoming Cryptography Standards, This Paper Summarizes Important Findings.

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