Simple method for realizingweil theorem in secure ECC generation

Abstract

How to quickly compute the number of points on an Elliptic Curve (EC) has been a longstanding challenge.The computational complexity of the algorithm usually employed makes it highly inefficient. Unlike the general EC, a simple method called the Weil theorem can be used to compute the order of an EC characterized by a small prime number, such as the Kobltiz EC characterized by two. The fifteen secure ECs recommended by the National Institute of Standards and Technology (NIST) Digital Signature Standard contain five Koblitz ECs whose maximum base domain reaches 571 bits. Experimental results show that the computation speed decreases for base domains exceeding 600 bits. In this paper, we propose a simple method that combines the Weil theorem with Pascals triangle, which greatly reduces the computational complexity. We have validated the performance of this method for base fields ranging from 2 100 to 2 1000 . Furthermore, this new method can be generalized to any ECs characterized by any small prime number.

Traffic clustering algorithm of urban data brain based on a hybrid-augmented architecture of quantum annealing and brain-inspired cognitive computing

“…In view of the defects of the current clustering algorithm, such as sensitivity to the initial cluster core, vulnerability to the training samples, and poor robustness, this study proposes a new hybrid-augmented architecture of QA and brain-inspired cognitive computing, which consists of QA, brain-inspired cognitive science, and classical computing. At present, QA and simulated annealing algorithms have made outstanding contributions in fields, such as cryptography [18][19][20] . Among these algorithms, QA, as the core algorithm of D-Wave machine, utilizes the quantum tunneling effects to present the tendency toward lowenergy states based on the quantum adiabatic computing theorem [21] with the potential to approximate, or even achieve the global optimum [22] , which can overcome the defect in which classical methods are easily trapped in the local optimal solution in large-scale cases.…”