Adding Points, Doubling Points: P, 2P, 3P, 4P, 8P etc.
Adding and doubling points is an essential element of the algebra of Elliptic curves.
Adding points involves joining the point P to the point Q, and finding the point R that the line between P and Q intersects, on E. As discussed in this article, if the line that passes through through P and Q is not vertical, it is guaranteed to pass through one other point on the elliptic curve, on the affine plane.
Finding 2P involves taking the tangent line to the curve (the point at which, in our addition process, Q = P, that is, we are now calculating P + Q = P + P = 2P ). We can then add P to 2P, etc. If P and Q are the same point (P = Q), the line is tangent to the curve. Again, if the line is not vertical, it will intersect with another point on the elliptic curve, on the affine plane.
Adding points to P repeatedly allows you to find P, 2P, 3P, 4P etc.
That is:
find the the tangent to the curve, the point that line intersects with E again, and that point's reflection, to find 2P
find the line that passes through P and 2P, the point that line intersects with E again, and that point's reflection, to fine 3P
find the line that passes through P and 3P, the point that line intersects with E again, and that point's reflection, to fine 4P
etc.
See animation below: different values for P, 2P, 3P, and 4P, for various starting values of P.
Doubling a point, repeatedly, requires taking the tangent to the curve, from P. Having used that to find the value for 2P (the additive inverse of R, where the tangent from P crosses the curve again), we then use 2P to find S = 2R = 4P, using the tangent at R in a similar fashion, and then T = 2S = 4R = 8P etc.
See animation below: different values for P, 2P, 4P etc. for various starting values of P, 2P = R, 2R = S, 2S = T, etc.
Point At Infinity
The point at infinity on an elliptic curve is the identity element of the curve's group structure, serving as the neutral element for the group operation of point addition. It acts as a distant, abstract point - where all lines parallel to the y-axis meet in the projective plane, commonly denoted as O.
Importantly, in terms of addition:
P + O = P
That is:
P + -P = P - P = O
Torsion Points
Torsion points on Elliptic Curves are special types of points that have a "finite order". In the context of Elliptic Curves, the order of a point is: the smallest positive integer n such that n times the point is the identity element, O (the "point at infinity").
A point P on an elliptic curve is called a torsion point if there exists a positive integer n such that:
nP = O
where P is the point on the curve, nP represents the point added to itself n times, and O is the identity element of the curve's group structure (the "point at infinity").
Properties of Torsion Points:
Finite Order: Torsion points have finite order, meaning there's a minimum positive integer n that results in reaching the identity element through repeated addition of the point to itself.
Subgroup Structure: The set of all torsion points on an elliptic curve forms a subgroup, known as the torsion subgroup of the curve. This subgroup has interesting algebraic and geometric properties.
Finite Fields
Torsion points are particularly important in cryptography, especially in the construction of elliptic curve cryptosystems. The security of these systems often relies on the properties of the torsion subgroup, especially its size and structure.
Theorem Application
The Mazur’s Torsion Theorem (also [1], [2]) provides a remarkable result regarding the torsion subgroups of elliptic curves over the rational numbers (ℚ). It states that the torsion subgroup of any elliptic curve over ℚ is isomorphic to one of fifteen possible groups, which are either cyclic groups of order n for n from 1 to 10 or 12, or a product of two cyclic groups with specific orders.
Torsion points serve as a key area of research in number theory and algebraic geometry, providing insights into the structure of elliptic curves and their applications in algorithms and cryptography.
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