science

Vectorial addition chain

Vectorial addition chain is a 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 Vectorial addition chain rather than just read about it. In short: In mathematics, for positive integers k and s, a vectorial addition chain is a sequence V of k-dimensional vectors vi of nonnegative integers for −k + 1 ≤ i ≤ s together with a sequence w, such that v−k+1 = [1, 0, 0, ..., 0, 0], v−k+2 = [0, 1, 0, ..., 0, 0], ⋮ ⋮ v0 = [0, 0, 0, ..., 0, 1], vi = vj + vr for all 1 ≤ i ≤ s with −k + 1 ≤ j, r ≤ i − 1, vs = [n0, ..., nk−1], w = (w1, ..., ws), wi = (j, r). For example, a v…

Key takeaways

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

Reference excerpt

v−k+1 = [1, 0, 0, ..., 0, 0], v−k+2 = [0, 1, 0, ..., 0, 0], ⋮ ⋮ v0 = [0, 0, 0, ..., 0, 1], vi = vj + vr for all 1 ≤ i ≤ s with −k + 1 ≤ j, r ≤ i − 1, vs = [n0, ..., nk−1], w = (w1, ..., ws), wi = (j, r). For example, a vectorial addition chain for [22, 18, 3] is

V = ([1, 0, 0], [0, 1, 0], [0, 0, 1], [1, 1, 0], [2, 2, 0], [4, 4, 0], [5, 4, 0], [10, 8, 0], [11, 9, 0], [11, 9, 1], [22, 18, 2], [22, 18, 3]) w = ((−2, −1), (1, 1), (2, 2), (−2, 3), (4, 4), (1, 5), (0, 6), (7, 7), (0, 8)) Vectorial addition chains are well suited to perform multi-exponentiation:

Input: Elements x0, ..., xk−1 of an abelian group G and a vectorial addition chain of dimension k computing [n0, ..., nk−1] Output: The element x0n0...xk−1nr−1 for i = −k + 1 to 0 do yi → xi+k−1 for i = 1 to s do yi → yj × yr return ys

Addition sequence An addition sequence for the set of integer S = {n0, ..., nr−1} is an addition chain v that contains every element of S. For example, an addition sequence computing

{47, 117, 343, 499} is

(1, 2, 4, 8, 10, 11, 18, 36, 47, 55, 91, 109, 117, 226, 343, 434, 489, 499). It is possible to find addition sequence from vectorial addition chains and conversely, so they are in a sense dual.

See also Addition chain Addition-chain exponentiation Exponentiation by squaring Non-adjacent form

References

Worked examples

Example 1 — a first encounter with Vectorial addition chain

Start with the simplest possible case. Write down what Vectorial addition chain claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In 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 Vectorial addition chain 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 Vectorial addition chain 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 Vectorial addition chain

In research: Vectorial addition chain appears in 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 Vectorial addition chain 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: Vectorial addition chain is common in secondary-school and first-year university syllabi. It links to neighbouring topics Addition chains, so understanding it makes those chapters shorter.
In everyday life: Look for Vectorial addition chain 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 Vectorial addition chain in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Vectorial addition chain 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 Vectorial addition chain out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Vectorial addition chain in simple terms?

In mathematics, for positive integers k and s, a vectorial addition chain is a sequence V of k-dimensional vectors vi of nonnegative integers for −k + 1 ≤ i ≤ s together with a sequence w, such that v−k+1 = [1, 0, 0, ..., 0, 0], v−k+2 = [0, 1, 0, ..., 0, 0], ⋮ ⋮ v0 = [0, 0, 0, ..., 0, 1], vi = vj +…

Why does Vectorial addition chain matter?

Because it connects several 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 Vectorial addition chain?

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 Vectorial addition chain.