Even with the above precautions, the benefits of elliptical curve cryptography over conventional RSA are widely accepted. Many experts are concerned that the mathematical algorithms behind RSA and Diffie-Hellman could be broken within 5 years, making ECC the only reasonable alternative. There are other representations of elliptical curves, but technically, an elliptical curve is the value of the equation that respond to an equation in two variables with grade two in one of the variables and three in the other. An elliptical curve is not only a pretty image, it also has some features that make it a good setting for cryptography. The improvement in ECDSA`s performance compared to the RSA is dramatic. Even with an older version of OpenSSL that has no elliptical curve code to optimize the assembly, an ECDSA signature with a 256-bit key is more than 20x faster than an RSA signature with a 2048-bit key. EC Diffie-Hellman Key Agreement Protocol: The Diffie-Hellman Key Memorandum of Understanding is the proposed basic public key cryptographic system for calculating key-sharing agreements for protocols with private and public keys. ECDH is the elliptical analog of the traditional diffie-hellman key chord algorithm. The Diffie-Hellman method requires no prior contact between the two parties. Each game generates a dynamic or ephemeral public key and a private key. They exchange their public keys. Each party then combines its private key with the public key of the other parties to calculate the common secret. We have shown that elliptical curves require less computing power, memory and communication bandwidth, giving it a clear lead over conventional cryptographic algorithms.
To date, elliptical curve cryptography is widely accepted compared to conventional cryptographic systems (DES, RSA, AES, etc.), which are generally performance-hungry, especially in wireless and portable devices. While ECC`s performance is impressive, the data security industry must ensure that the security system has been thoroughly reviewed in the public forum using the elliptical curve algorithm and has also been specified by major standards around the world. But we think that the cryptography of elliptical curves is here today and is undoubtedly the next generation of public key cryptography of choice. A crypto-system of elliptical curves can be defined by selecting a prime number as maximum, a curve equation and a public point on the curve. A private key is a private number, and a public key is the public point marked with itself private periods. The calculation of the private key from the public key in this type of cryptographic system is called logarithm function of elliptical-discrete curves. This turns out to be the Trapdoor feature we were looking for. Implementation of multiplication/addition via a group of modulo p elliptical curves: Consider an equation of the shape Q-kP. For a whole positive number, we have the curve multiplication map designated [k] to itself. This card takes a P point at P…..-P (k summands).
The notation [k] is extended to k≤0 by defining P-O and [-k]P- ([k]P). For exampleP – P-P, P – P-P-P and [-3]P – -(P-P-P). In addition, for a particular P point on an elliptical curve E, there is a minimum positive n whole number, so that nP-O, the ID point or infinity point. The entire number is called point P order. It is known that n is a divider of the order of the E curve.