computer science

Table of costs of operations in elliptic curves

Table of costs of operations in elliptic curves 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 Table of costs of operations in elliptic curves rather than just read about it. In short: Elliptic curve cryptography is a popular form of public key encryption that is based on the mathematical theory of elliptic curves. Points on an elliptic curve can be added and form a group under this addition operation.

Key takeaways

Table of costs of operations in elliptic curves 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 Table of costs of operations in elliptic curves to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Table of costs of operations in elliptic curves from memory before moving on to harder problems.

Reference excerpt

Elliptic curve cryptography is a popular form of public key encryption that is based on the mathematical theory of elliptic curves. Points on an elliptic curve can be added and form a group under this addition operation. This article describes the computational costs for this group addition and certain related operations that are used in elliptic curve cryptography algorithms.

Abbreviations for the operations The next section presents a table of all the time-costs of some of the possible operations in elliptic curves. The columns of the table are labelled by various computational operations. The rows of the table are for different models of elliptic curves. These are the operations considered:

To see how adding (ADD) and doubling (DBL) points on elliptic curves are defined, see The group law. The importance of doubling to speed scalar multiplication is discussed after the table. For information about other possible operations on elliptic curves see http://hyperelliptic.org/EFD/g1p/index.html.

Tabulation Under different assumptions on the multiplication, addition, inversion for the elements in some fixed field, the time-cost of these operations varies. In this table it is assumed that:

I = 100M, S = 1M, ×param = 0M, add = 0M, ×const = 0M This means that 100 multiplications (M) are required to invert (I) an element; one multiplication is required to compute the square (S) of an element; no multiplication is needed to multiply an element by a parameter (×param), by a constant (×const), or to add two elements. For more information about other results obtained with different assumptions, see http://hyperelliptic.org/EFD/g1p/index.html

Importance of doubling In some applications of elliptic curve cryptography and the elliptic curve method of factorization (ECM) it is necessary to consider the scalar multiplication [n]P. One way to do this is to compute successively:

P , [ 2 ] P = P + P , [ 3 ] P = [ 2 ] P + P , … , [ n ] P = [ n − 1 ] P + P {\displaystyle P,\quad [2]P=P+P,\quad [3]P=[2]P+P,\quad \dots ,\quad [n]P=[n-1]P+P}

But it is faster to use double-and-add method; for example, [5]P = [2]([2]P) + P. In general, to compute [k]P, write

k = ∑ i ≤ ℓ k i 2 i {\displaystyle k=\sum _{i\leq \ell }k_{i}2^{i}}

with ki ∈ {0,1} and ℓ = ⌊log2 k⌋ and kℓ = 1, then:

[ 2 ] ( . . . . ( [ 2 ] ( [ 2 ] ( [ 2 ] ( [ 2 ] ( [ 2 ] P + [ k ( l − 1 ) ] P ) + [ k ( l − 2 ) ] P ) + [ k ( l − 3 ) ] P ) + … ) ⋯ + [ k 1 ] P ) + [ k 0 ] P = [ 2 l ] P + [ k ( l − 1 ) 2 l − 1 ] P + ⋯ + [ k 1 2 ] P + [ k 0 ] P . {\displaystyle {\begin{aligned}&[2](....([2]([2]([2]([2]([2]P+[k_{(l-1)}]P)+[k_{(l-2)}]P)+[k_{(l-3)}]P)+\dots )\dots +[k_{1}]P)+[k_{0}]P\\&\qquad =[2^{l}]P+[k_{(l-1)}2^{l-1}]P+\dots +[k_{1}2]P+[k_{0}]P\end{aligned}}.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Table of costs of operations in elliptic curves

Start with the simplest possible case. Write down what Table of costs of operations in elliptic curves 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 Table of costs of operations in elliptic curves 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 Table of costs of operations in elliptic curves 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 Table of costs of operations in elliptic curves

In research: Table of costs of operations in elliptic curves 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 Table of costs of operations in elliptic curves 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: Table of costs of operations in elliptic curves is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational number theory, Cryptographic attacks, Elliptic curve cryptography, so understanding it makes those chapters shorter.
In everyday life: Look for Table of costs of operations in elliptic curves 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 Table of costs of operations in elliptic curves in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Table of costs of operations in elliptic curves 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.
Explain Table of costs of operations in elliptic curves out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Table of costs of operations in elliptic curves in simple terms?

Why does Table of costs of operations in elliptic curves 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 Table of costs of operations in elliptic curves?

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 Table of costs of operations in elliptic curves.