computer science

Itoh–Tsujii inversion algorithm

Itoh–Tsujii inversion algorithm 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 Itoh–Tsujii inversion algorithm rather than just read about it. In short: While the algorithm is often called the Itoh-Tsujii algorithm, it was first presented by Gui-Liang Feng. Feng's paper was received on March 13, 1987 and published in October 1989.

Key takeaways

Itoh–Tsujii inversion algorithm 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 Itoh–Tsujii inversion algorithm to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Itoh–Tsujii inversion algorithm from memory before moving on to harder problems.

Reference excerpt

While the algorithm is often called the Itoh-Tsujii algorithm, it was first presented by Gui-Liang Feng. Feng's paper was received on March 13, 1987 and published in October 1989. Itoh and Tsujii's paper was received on July 8, 1987 and published in 1988. Feng and Itoh-Tsujii algorithm is first used to invert elements in finite field GF(2m) using the normal basis representation of elements, however, it is generic and can be used for other bases, such as the polynomial basis. It can also be used in any finite field GF(pm). The algorithm is as follows:

Input: A ∈ GF(pm) Output: A−1 r ← (pm − 1)/(p − 1) compute Ar−1 in GF(pm) compute Ar = Ar−1 · A compute (Ar)−1 in GF(p) compute A−1 = (Ar)−1 · Ar−1 return A−1 This algorithm is fast because steps 3 and 5 both involve operations in the subfield GF(p). Similarly, if a small value of p is used, a lookup table can be used for inversion in step 4. The majority of time spent in this algorithm is in step 2, the first exponentiation. This is one reason why this algorithm is well suited for the normal basis, since squaring and exponentiation are relatively easy in that basis. This algorithm is based on the fact that GF(pm)* is a cyclic group of order pm-1. Given a nonzero element A in finite field GF(2m), we have

A − 1 = A 2 m − 2 = A 2 m − 1 ⋅ A 2 m − 2 ⋅ A 2 m − 3 ⋯ A 2 2 ⋅ A 2 1 = ∏ i = 1 m − 1 A 2 i . {\displaystyle A^{-1}=A^{2^{m}-2}=A^{2^{m-1}}\cdot A^{2^{m-2}}\cdot A^{2^{m-3}}\cdots A^{2^{2}}\cdot A^{2^{1}}=\prod _{i=1}^{m-1}{A^{2^{i}}}.}

The above A−1 expression itself is close to that of the multiplicative Norm function in finite field, which is defined as

N o r m ( A ) = ∏ i = 0 m − 1 A 2 i . {\displaystyle Norm(A)=\prod _{i=0}^{m-1}{A^{2^{i}}}.}

This viewpoint leads us to consider the additive absolute Trace function

, which is defined as

T r ( A ) = ∑ i = 0 m − 1 A 2 i . {\displaystyle Tr(A)=\sum _{i=0}^{m-1}{A^{2^{i}}}.}

If Tr(A)=0, then we have

A = ∑ i = 1 m − 1 A 2 i {\displaystyle A=\sum _{i=1}^{m-1}{A^{2^{i}}}}

and can express A−1 as

A − 1 = A − 2 ∑ i = 1 m − 1 A 2 i = ∑ i = 1 m − 1 A 2 i − 2 = ∑ j = 0 m − 2 ( A 2 ) 2 j − 1 . {\displaystyle A^{-1}=A^{-2}\sum _{i=1}^{m-1}{A^{2^{i}}}=\sum _{i=1}^{m-1}{A^{2^{i}-2}}=\sum _{j=0}^{m-2}{(A^{2})^{2^{j}-1}}.}

In some GF(2m)s, for example, GF(28) used in Advanced Encryption Standard (AES), this formula needs 1 less multiplication operation than Feng and Itoh-Tsujii algorithm for elements with Trace value 0: because

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Itoh–Tsujii inversion algorithm

Start with the simplest possible case. Write down what Itoh–Tsujii inversion algorithm 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 Itoh–Tsujii inversion algorithm 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 Itoh–Tsujii inversion algorithm 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 Itoh–Tsujii inversion algorithm

In research: Itoh–Tsujii inversion algorithm 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 Itoh–Tsujii inversion algorithm 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: Itoh–Tsujii inversion algorithm is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational number theory, Finite fields, so understanding it makes those chapters shorter.
In everyday life: Look for Itoh–Tsujii inversion algorithm 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 Itoh–Tsujii inversion algorithm in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Itoh–Tsujii inversion algorithm 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 Itoh–Tsujii inversion algorithm out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Itoh–Tsujii inversion algorithm in simple terms?

While the algorithm is often called the Itoh-Tsujii algorithm, it was first presented by Gui-Liang Feng. Feng's paper was received on March 13, 1987 and published in October 1989.

Why does Itoh–Tsujii inversion algorithm 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 Itoh–Tsujii inversion algorithm?

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 Itoh–Tsujii inversion algorithm.