EC Cryptography Tutorials - Herong's Tutorial Examples

∟Reduced Elliptic Curve Groups

∟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, E_p(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.

Table of Contents

 About This Book

 Geometric Introduction to Elliptic Curves

 Algebraic Introduction to Elliptic Curves

 Abelian Group and Elliptic Curves

 Discrete Logarithm Problem (DLP)

 Finite Fields

 Generators and Cyclic Subgroups

►Reduced Elliptic Curve Groups

 Converting Elliptic Curve Groups

 Elliptic Curves in Integer Space

 Python Program for Integer Elliptic Curves

 Elliptic Curves Reduced by Modular Arithmetic

 Python Program for Reduced Elliptic Curves

 Point Pattern of Reduced Elliptic Curves

 Integer Points of First Region as Element Set

 Reduced Point Additive Operation

 Modular Arithmetic Reduction on Rational Numbers

 Reduced Point Additive Operation Improved

 What Is Reduced Elliptic Curve Group

 Reduced Elliptic Curve Group - E23(1,4)

 Reduced Elliptic Curve Group - E97(-1,1)

 Reduced Elliptic Curve Group - E127(-1,3)

 Reduced Elliptic Curve Group - E1931(443,1045)

►What Is Hasse's Theorem

 Finite Elliptic Curve Group, Eq(a,b), q = p^n

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 EC Cryptography in Java

 Standard Elliptic Curves

 Terminology

 References

 Full Version in PDF/EPUB

What Is Hasse's Theorem - Updated in 2022, by Dr. Herong Yang