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An important aspect in the study of elliptic curves is devising effective ways of counting points on the curve.There have been several approaches to do so, and the algorithms devised have proved to be useful tools in the study of various fields such as number theory, and more recently in cryptography and Digital Signature Authentication (See elliptic curve cryptography and elliptic curve DSA).
If the instantiated polynomial (, ()) has a root (′) in then is an Elkies prime, and we may compute a polynomial () whose roots correspond to points in the kernel of the -isogeny from to ′. The polynomial f l {\displaystyle f_{l}} is a divisor of the corresponding division polynomial used in Schoof's algorithm, and it has significantly ...
Schoof's algorithm is an efficient algorithm to count points on elliptic curves over finite fields.The algorithm has applications in elliptic curve cryptography where it is important to know the number of points to judge the difficulty of solving the discrete logarithm problem in the group of points on an elliptic curve.
Hasse's theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic curve over a finite field, bounding the value both above and below. If N is the number of points on the elliptic curve E over a finite field with q elements, then Hasse's result states that
A minimum of 300 representative points (preferably 500 points) should be used to perform the count. On each randomly selected point that lands on a sand grain, the operator must determine the make-up of the area chosen, i.e. whether it is a mineral grain that is sand sized (larger than 62.5 micrometers) or a finer-grained fragment of another ...
Let X be a Riemann surface.Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function :), we can associate a differential form of compact support, the Poincaré dual of c, with the property that integrals along c can be calculated by integrals over X:
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Averages per game are denoted by *PG, e.g. PPG (points), BLKPG or BPG (blocks), STPG or SPG (steals), APG (assists), RPG (rebounds) and MPG (minutes).Sometime the players statistics are divided by minutes played and multiplied by 48 minutes (had he played the entire game), denoted by * per 48 min. or *48M.