computer science

Tripling-oriented Doche–Icart–Kohel curve

Tripling-oriented Doche–Icart–Kohel curve is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Tripling-oriented Doche–Icart–Kohel curve rather than just read about it. In short: The tripling-oriented Doche–Icart–Kohel curve is a form of an elliptic curve that has been used lately in cryptography; it is a particular type of Weierstrass curve. At certain conditions some operations, as adding, doubling or tripling points, are faster to compute using this form.

Key takeaways

Tripling-oriented Doche–Icart–Kohel curve belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Tripling-oriented Doche–Icart–Kohel curve to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Tripling-oriented Doche–Icart–Kohel curve from memory before moving on to harder problems.

Reference excerpt

The tripling-oriented Doche–Icart–Kohel curve is a form of an elliptic curve that has been used lately in cryptography; it is a particular type of Weierstrass curve. At certain conditions some operations, as adding, doubling or tripling points, are faster to compute using this form. The tripling-oriented Doche–Icart–Kohel curve, often called with the abbreviation 3DIK was introduced by Christophe Doche, Thomas Icart, and David R. Kohel in.

Definition

Let K {\displaystyle K} be a field of characteristic different form 2 and 3. An elliptic curve in tripling oriented Doche–Icart–Kohel form is defined by the equation:

T a : y 2 = x 3 + 3 a ( x + 1 ) 2 {\displaystyle T_{a}\ :\ y^{2}=x^{3}+3a(x+1)^{2}}

with a ∈ K {\displaystyle a\in K} . A general point P on T a {\displaystyle T_{a}} has affine coordinates ( x , y ) {\displaystyle (x,y)} . The "point at infinity" represents the neutral element for the group law and it is written in projective coordinates as O = (0:1:0). The negation of a point P = (x, y) with respect to this neutral element is −P = (x, −y).

The group law Consider an elliptic curve in the Tripling-oriented Doche-Icart-Kohel form in affine coordinates:

T a : y 2 = x 3 + 3 a ( x + 1 ) 2 , a ≠ 0 , 9 4 . {\displaystyle T_{a}:\quad y^{2}=x^{3}+3a(x+1)^{2},\qquad a\neq 0,{\tfrac {9}{4}}.}

As in other forms of elliptic curves, it is possible to define some "operations" between points, such as adding points, or doubling (See also The group law). In the following sections formulas to add, negate and doubling points are given. The addition and doubling formulas are often used for other operations: given a point P on an elliptic curve it is possible to compute [n]P, where n is an integer, using addition and doubling; computing multiples of points is important in elliptic curve cryptography and in Lenstra elliptic curve factorization.

Addition Given P 1 = ( x 1 , y 1 ) {\displaystyle P_{1}=(x_{1},y_{1})} and P 2 = ( x 2 , y 2 ) {\displaystyle P_{2}=(x_{2},y_{2})} on T a {\displaystyle T_{a}} , the point P 3 = ( x 3 , y 3 ) = P 1 + P 2 {\displaystyle P_{3}=(x_{3},y_{3})=P_{1}+P_{2}} has coordinates:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Tripling-oriented Doche–Icart–Kohel curve

Start with the simplest possible case. Write down what Tripling-oriented Doche–Icart–Kohel curve claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Tripling-oriented Doche–Icart–Kohel curve before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Tripling-oriented Doche–Icart–Kohel curve ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Tripling-oriented Doche–Icart–Kohel curve

In research: Tripling-oriented Doche–Icart–Kohel curve appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Tripling-oriented Doche–Icart–Kohel curve in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Tripling-oriented Doche–Icart–Kohel curve is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elliptic curve cryptography, Elliptic curves, so understanding it makes those chapters shorter.
In everyday life: Look for Tripling-oriented Doche–Icart–Kohel curve outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Tripling-oriented Doche–Icart–Kohel curve in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Tripling-oriented Doche–Icart–Kohel curve means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
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Frequently asked questions

What is Tripling-oriented Doche–Icart–Kohel curve in simple terms?

Why does Tripling-oriented Doche–Icart–Kohel curve matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Tripling-oriented Doche–Icart–Kohel curve?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Tripling-oriented Doche–Icart–Kohel curve.