What Is Hasse's Theorem

This section describes Hasse's Theorem, which states that the order, n, of a reduced elliptic curve group, Ep(a,b), is bounded in the range of [p+1 - 2*sqrt(p), p+1 + 2*sqrt(p)].

What Is Hasse's Theorem? Hasse's Theorem is also called Hasse Bound, which provides an estimate of the number of points on an elliptic curve over a finite field, bounding the value both above and below.

```For a given elliptic curve E(a,b) over a finite field with q elements,
the number of points, n, on the curve satisfies the following condition:
|n - (q+1)| <= 2*sqrt(q)
```

If we apply Hasse's Theorem to our reduced elliptic curve group definition, Ep(a,b)), we have:

```Given reduced elliptic curve group Ep(a,b),
the group order, n, satisfies the following condition:
|n - (p+1)| <= 2*sqrt(p)

Or n is in a range as expressed below:
p+1 - 2*sqrt(p) <= n <= p+1 + 2*sqrt(p)
```

It's interesting to see that the group order is only depending on the prime number used in the modular arithmetic reduction, not on coefficients, a and b, of the elliptic curve equation.

Hasse's Theorem is named after German mathematician Helmut Hasse (25 August 1898 - 26 December 1979):

For more details, see "Hasse's theorem on elliptic curves" at en.wikipedia.org/wiki/Hasse%27s_theorem_on_elliptic_curves.

Last update: 2019.