Design and Analysis of a Novel Elliptic Curve for Cryptography Applications
Nalini S · 2024
Elliptic Curve Cryptography (ECC) is a highly effective and efficient subset of public-key cryptography, utilizing the unique mathematical attributes of elliptic curves to facilitate secure communication across potentially compromised channels.The strength and efficiency of ECC largely stem from the complexity of the Elliptic Curve Discrete Logarithm Problem (ECDLP), which makes it extremely difficult, even with advanced computational power, to deduce the private key from the corresponding public key [12].This efficiency allows ECC to achieve comparable security to other cryptography systems, such as RSA, but with significantly smaller key sizes, resulting in faster computations and reduced storage and transmission requirements.Despite the significant advancements in ECC, there remains an ongoing need for the development of new elliptic curves that can provide enhanced security and efficiency.Current standardized curves, such as Curve25519 and P-256, have been extensively analyzed and are widely used, but they may not fully leverage recent advances in mathematical techniques and computational optimization.Additionally, the increasing computational power and emerging quantum computing technologies pose potential threats to the long-term security of existing cryptography systems. A. Objectives:This research aims to develop an innovative elliptic curve characterized by advanced algebraic structures that ensure both heightened security and computational efficiency.The key objectives include: Curve Design: Identifying suitable prime fields and parameters to construct a new elliptic curve with optimal properties.Mathematical Analysis: Evaluating the mathematical foundations and properties of the curve, including order, embedding degree, and resistance to known attacks.3. Performance Evaluation: Assessing the efficiency of the curve in terms of scalar multiplication speed and comparing it with existing standardized curves.4. Security Implications: Analyzing the theoretical and practical security aspects of the curve to ensure its robustness against contemporary and future cryptography threats.