Abstract: The present invention comprises fast new methods for computing high-precision solutions of Frobenius equations that arise in elliptic-curve cryptography. In particular, this invention may be used to accelerate the computation of the number of points on an elliptic curve over a finite field. The advantage over methods in prior art is that the invention is faster than previously known methods. The methods enable optimally fast canonical lifting of elliptic curves defined over finite fields, optimally fast pre-computations to determine an efficient representation of intermediate quantities, and optimally fast lifting of finite-field elements to compute multiplicative representatives. Furthermore the invention enables rapid computation of norms and traces amongst other applications.

Abstract: Methods for determining whether an arbitrary elliptic curve over a binary field is secure, by using a novel non-converging Arithmetic-Geometric Mean iteration to determine the exact number of points on the curve. The methods provide rapid generation of secure curves for Elliptic-Curve Cryptography by selecting a secure curve from among candidate curves with the new method. The secure curve chosen is a curve whose number of points, is found to be divisible by a large prime number. The number of points on candidate curves is computed by a first phase, which lifts the curve to a certain related curve, followed by a second phase, which computes a certain norm that yields the result. The new Arithmetic-Geometric Mean iteration is used for the lifting phase or for the norm phase or for both.

Abstract: The present invention comprises fast new methods for computing high-precision solutions of Frobenius equations that arise in elliptic-curve cryptography. In particular, this invention may be used to accelerate the computation of the number of points on an elliptic curve over a finite field. The advantage over methods in prior art is that the invention is faster than previously known methods. The methods enable optimally fast canonical lifting of elliptic curves defined over finite fields, optimally fast pre-computations to determine an efficient representation of intermediate quantities, and optimally fast lifting of finite-field elements to compute multiplicative representatives. Furthermore the invention enables rapid computation of norms and traces amongst other applications.

Abstract: The present invention is a fast new method for determining whether an arbitrary elliptic curve over a binary field is secure, by using a novel non-converging Arithmetic-Geometric Mean iteration to determine the exact number of points on the curve. This invention is used for the rapid generation of secure curves for Elliptic-Curve Cryptography by selecting a secure curve from among candidate curves with the new method. The secure curve chosen is a curve whose number of points, determined using the invention, is found to be divisible by a large prime number. The number of points on candidate curves is computed by a first phase, which lifts the curve to a certain related curve, followed by a second phase, which computes a certain norm that yields the result. The new Arithmetic-Geometric Mean iteration is used for the lifting phase or for the norm phase or for both.