Is there any body who knows what is the application of Hasse-Weil theorem in elliptic curves and cryptography? Any help would be great thanks....
2026-04-03 00:00:56.1775174456
The applications of Hasse-weil theorem
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The Hasse-Weil theorem is a generalized version of Hasse's theorem for other algebraic curves. Hasse's Theorem is used for Elliptic Curves defined over finite fields in determining upper and lower bounds for the group order. You only need to know the order of the finite field to do so.
If the order of the field is $q$ and the group order (number of points in the group) is $N$,
$$q+1-2{\sqrt q} ≤ N ≤ q+1+2{\sqrt q}$$
In terms of applications of this, this is especially useful in Shanks' Baby-Step Giant-Step method of solving the Elliptic Curve Discrete Logarithm problem, as it makes it relatively easy to estimate the group order. Although other methods exist to find the exact group order (Schoof's Algorithm), this is less complex.