I have a new method, so the comparison among methods play a crucial role in improving this method.
Try to look at the index calculus. I dont know if it is the best but it seems to be enough efficient
What about Pollard's Rho method and its parallelized? it may be more efficient and fast.
We have also the Pohlig-Hellman Method. You may look the thesis of Matthew Musson it's powerful with remarkable special attacks (specially the Weil Descent and the GHS Attack)
For the generic curve over the prime feild, the best method is Pollard's Rho and its parallelized.
It depends on the type of curve. For Koblitz curve or any pairing-friendly curve still, Parallel implementation of Pollard's Rho would be the best choice. For implementation perspective, you can read this Kajitani et al's work https://www.researchgate.net/publication/319217358_Web-based_Volunteer_Computing_for_Solving_the_Elliptic_Curve_Discrete_Logarithm_Problem_Web-based_Volunteer_Computing_for_Solving_the_ECDLP
Article Web-based Volunteer Computing for Solving the Elliptic Curve...
As of now, the most efficient method for solving the Elliptic Curve Discrete Logarithm Problem (ECDLP) is the Pollard's rho algorithm, specifically the variant known as the Pollard's rho algorithm with kangaroos. This algorithm is widely used for attacking the ECDLP on elliptic curves.
The reason why the Pollard's rho algorithm with kangaroos is considered efficient for solving the ECDLP on elliptic curves is because it takes advantage of the structure of elliptic curves and the properties of the group operations involved. This algorithm is a probabilistic algorithm that uses a random walk approach to find the discrete logarithm efficiently.
The Pollard's rho algorithm with kangaroos has been extensively studied and optimized over the years, leading to its effectiveness in solving the ECDLP. It is able to handle large prime fields and elliptic curves efficiently, making it a popular choice for attacking ECDLP in practice.
It's important to note that the security of elliptic curve cryptography relies on the difficulty of solving the ECDLP, and the efficiency of attacking methods like the Pollard's rho algorithm with kangaroos is a crucial factor in evaluating the strength of elliptic curve cryptosystems.
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