Mathematics, Cryptography, Quantum on kenji.blog

[Complete Guide] What is a Quantum Computer? ~The Ultimate Computing Principle from Scratch~

Sat, 05 Sep 2026 22:10:00 +0900

Introduction: The “Paradigm Shift in Computing” Brought by Quantum Computers

In recent years, not a day goes by without seeing the term “quantum computer” in the news or technical articles. Sci-fi-like stories are often told, such as “calculations that would take current supercomputers thousands of years can be finished in minutes” or “all current cryptographic technologies might be broken.” Giant IT companies like Google, IBM, and Microsoft, as well as universities and startups worldwide, are fiercely competing to commercialize this dream technology.

However, if asked, “What exactly is a quantum computer?”, few people could answer accurately. Many have a vague image of a “magic box that can calculate all combinations simultaneously,” but strictly speaking, that is incorrect.

In this article, we will thoroughly explain from the basics how quantum computers fundamentally differ from classical computers (the PCs and smartphones we usually use), and how they utilize mysterious phenomena of quantum mechanics such as “Superposition,” “Entanglement,” and “Quantum gates” for calculations, in a professional yet easy-to-understand manner. By the time you finish reading this article, you should clearly understand the essential greatness of quantum computers and their current challenges.

Chapter 1: The Crucial Difference Between Classical and Quantum Computers

To understand how a quantum computer works, we first need to review how the “classical computers” we currently use operate.

Comparison Table: Classical Computers vs Quantum Computers

Item	Classical Computer	Quantum Computer
Basic Unit	Bit (0 or 1)	Qubit (superposition of 0 and 1)
State Representation	Deterministic	Probabilistic (undetermined until observed)
Calculation Method	Sequential processing (requires physical cores for parallelization)	Quantum parallelism (manipulates exponential states simultaneously)
Excels at	Arithmetic operations, everyday data processing	Prime factorization, quantum chemistry calculations
Error Tolerance	Very strong	Very weak (requires cryogenic environments and error correction)

The World of Classical Computers: The “Bit” of 0 or 1

Classical computers represent all information in a state of either “0” or “1”. This is called a Bit. Physically, it is represented by whether the voltage of a transistor on a semiconductor chip is high (1) or low (0). The high-quality photos on your smartphone, the text you are reading now, and your favorite YouTube videos are all ultimately reduced to a massive sequence of 0s and 1s. Computation is nothing more than the process of applying operations to this sequence of 0s and 1s by combining basic logic circuits such as AND, OR, and NOT. This is a very reliable and deterministic world. If the input is the same, you will always get the same output.

The World of Quantum Computers: The “Qubit” That is Both 0 and 1

On the other hand, the smallest unit of information in a quantum computer is called a Qubit (Quantum bit). The biggest feature of a qubit is that, unlike a classical bit which is strictly either “0” or “1”, it can take on a state where “0 and 1 are mixed with specific probabilities.” This is called “Superposition”.

For example, if a classical bit is a coin placed with either “heads” or “tails” facing up, a qubit is often compared to a “coin constantly spinning in the air.” A spinning coin cannot be said to be heads or tails; both states are superimposed. And the moment the coin falls to the floor and stops moving (in quantum mechanics, this is called “observation”), it is determined as either “heads” or “tails” for the first time.

Quantum computers incorporate this very property unique to the microscopic world (quantum mechanics)—where “the state is not determined until observed”—directly into the information processing process.

Chapter 2: Three Quantum Mechanical Properties That Fundamentally Change Computing

The source of the astonishing computing power of quantum computers is not simply a high clock frequency or small components. It lies in utilizing the laws of physics themselves as computational resources. The following three quantum mechanical phenomena are the keys.

1. Superposition and Exponential Information Capacity

As mentioned earlier, a qubit can simultaneously hold the states of both 0 and 1. One qubit is a “superposition of 0 and 1,” but what happens when you increase the number of qubits?

1 Qubit: Superposition of 2 states (0, 1)
2 Qubits: Superposition of 4 states (00, 01, 10, 11)
3 Qubits: Superposition of 8 states
N Qubits: Superposition of $2^N$ patterns

With just 50 qubits, you can simultaneously hold $2^{50}$ (about 1.1 quadrillion) states. And with just 300 qubits, you can hold $2^{300}$ patterns at once—a number greater than all the atoms in the universe! This exponential information holding capacity is the foundation of a quantum computer’s potential. It is physically impossible for a classical computer to store more states in memory than there are atoms in the universe.

2. Entanglement: Spooky Action at a Distance

Quantum entanglement is such a counter-intuitive and mysterious phenomenon that Albert Einstein called it “Spooky action at a distance” and refused to accept it throughout his life.

When multiple qubits enter a state of “quantum entanglement,” they become strongly linked to each other, forming a shared destiny: “when the state of one is determined, the state of the other is instantly determined, no matter how far apart they are.”

For example, suppose we have two entangled qubits A and B (each in a superposition of 0 and 1). If we observe A and find it to be “0,” the state of B is instantly determined (for example, to definitely be “1”), exceeding the speed of light, which is the limit of information transmission. In quantum computers, this entanglement is used to represent complex correlations between multiple qubits and perform massively parallel information processing. Without entanglement, the computing power of quantum computers would not be much different from that of classical computers.

3. Quantum Interference: The Magic That Brings Out the Right Answer

You might think, “If it can hold all patterns simultaneously, couldn’t it calculate them all at once in parallel and output the answer instantly?” This is the most common misconception about quantum computers. Even if you calculate in a superposition state, you must “observe” to finally know the answer. However, the moment you observe, the state randomly collapses into one of the $2^N$ patterns. This would just give you a random answer.

Here is where “Quantum Interference” comes in. When waves collide, they strengthen each other where their wavelengths match, and cancel each other out where they are out of sync (the principle is essentially the same as noise-canceling headphones).

An excellent “quantum algorithm” masterfully manipulates quantum states during the computation process so that “the probability amplitudes of states (waves) leading to the correct answer amplify each other” and “the probability amplitudes of states leading to incorrect answers cancel each other out.” It is designed so that when ultimately observed, the “correct answer” pops out with a probability as close to 100% as possible. Designing this interference process well is the very essence of quantum programming.

Chapter 3: How Do They Compute? “Quantum Gates” and “Quantum Circuits”

Just as classical computers use logic gates (AND, OR, NOT, etc.) to perform calculations, quantum computers apply operations called “Quantum Gates” to qubits to advance calculations. A combination of multiple quantum gates is called a ** Quantum Circuit**.

The state of a qubit is mathematically represented as a point on the surface of a 3-dimensional sphere called a “Bloch sphere.” The North Pole is “0”, the South Pole is “1”, and the equator represents a state where “0 and 1 are equally superimposed.” A quantum gate is nothing more than an operation that rotates the state (vector) on the surface of this sphere.

Let’s introduce some representative quantum gates.

1. Hadamard Gate (H Gate)

This is the most fundamental gate unique to quantum computers, not found in classical computers. When a qubit exactly in the “0” state passes through an H gate, it creates a “perfect superposition state” (a point on the equator of the Bloch sphere) where 0 and 1 are observed with exactly 50% probability each. As an initialization step for quantum computation, many algorithms begin by applying this H gate to all qubits.

2. Pauli Gates (X, Y, Z Gates)

These gates correspond to the NOT gate (flipping 0 to 1, and 1 to 0) of classical computers. On the Bloch sphere, they correspond to 180-degree rotations around the X, Y, and Z axes. The X gate, in particular, flips the North Pole (0) to the South Pole (1), working exactly the same as a classical NOT gate. The Z gate has the role of flipping the “phase” (like the timing of a wave) of a superposition, which is extremely important for causing quantum interference.

3. CNOT Gate (Controlled NOT Gate)

This is a hyper-important gate for creating quantum entanglement. It uses two qubits (a control bit and a target bit). It operates as follows: “If the control bit is 1, flip the state of the target bit (apply an X gate). If the control bit is 0, do nothing.” At first glance, it looks like a simple IF conditional branch, but what happens if the control bit is in a “superposition state of 0 and 1”? The target bit becomes a “superposition of flipped and unflipped states,” and the destinies of the two bits become completely linked. The two qubits beautifully become “entangled.”

By placing and applying these gates in sequence from left to right like a musical score, complex algorithms are executed.

Chapter 4: What Are Quantum Computers Good At, and What Are They Bad At?

Here is an important fact: quantum computers are not omnipotent gods. For everyday tasks like web browsing, video rendering, Excel macro processing, or running general smartphone apps, quantum computers will probably never surpass classical computers. Classical computers, which are already highly optimized and boast overwhelming speed and low cost, are better suited for these sequential processes.

Quantum computers truly shine only when applied to “specific problems where the combinations of calculations explode exponentially for classical computers, taking time comparable to the age of the universe.” This is called “Quantum Supremacy” or “Quantum Advantage.”

What Quantum Computers Excel At (Killer Applications)

1. Prime Factorization and Cryptanalysis (Shor’s Algorithm)

Currently, secure communications on the Internet (such as credit card payments and sending personal information) are protected by “RSA encryption,” which is based on the premise that “prime factorization of huge numbers is practically impossible (takes an enormous amount of time) for classical computers.” However, by using “Shor’s Algorithm” discovered by mathematician Peter Shor in 1994, a quantum computer can skillfully use interference to solve this at a dramatic speed (polynomial time). Because of this, there is a risk that the current cryptographic systems will collapse in the future, and central banks and government agencies worldwide are rushing to transition to “Post-Quantum Cryptography.”

2. Quantum Chemistry Calculations, New Materials, and Drug Discovery

The behavior of molecules and atoms in the natural world originally follows the laws of quantum mechanics. When trying to simulate the behavior of complex molecules on a classical computer, the combinations of electron interactions explode, hitting the computational limits even for relatively small molecules. As Nobel laureate Richard Feynman said, “Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical,” quantum computers exert overwhelming native power in material simulations. Breakthroughs that solve humanity’s challenges are expected, such as the design of revolutionary new drugs, the discovery of room-temperature superconductors, the development of highly efficient solar cells and battery materials, and the synthesis of energy-efficient fertilizers.

3. Combinatorial Optimization Problems and Search (Grover’s Algorithm)

Quantum algorithms also demonstrate power for problems that involve finding the optimal choice from an enormous number of options (logistics route optimization, financial portfolio optimization, etc.). By using “Grover’s Algorithm,” the target data can be found from an unsorted database in the square root of the time it would take a classical computer. For example, if there are 100 million pieces of data, a search that takes a maximum of 100 million tries classically can be completed in just about 10,000 tries.

Chapter 5: The Hardware Walls In the Way: “Decoherence” and “Quantum Error Correction”

Although theoretically magically powerful, the path to practical application of quantum computers is blocked by extremely high and steep physical walls. The biggest enemy is “Noise”.

The “superposition” and “quantum entanglement” of qubits are extremely delicate and fragile states. Even the slightest ambient heat, electromagnetic fluctuation, or cosmic rays can instantly cause that magical state to collapse, turning it into just a classical bit. This phenomenon is called “Decoherence”.

The Fierce Competition of Physical Implementation Methods

Currently, there is a battle for supremacy worldwide regarding how to physically create these delicate qubits.

Superconducting Method: Adopted by Google, IBM, Amazon, etc. It uses loop-shaped superconducting circuits and controls the quantum state by cooling it to a cryogenic temperature close to absolute zero (about -273°C) using massive refrigerators. It is currently the most advanced method and the easiest to scale up the number of qubits, but the cooling equipment is huge and expensive.
Trapped Ion Method: Adopted by IonQ, Quantinuum, etc. It traps ions (atoms) in a vacuum using electromagnetic fields and controls them by shining precise lasers. Its strength is that all qubits are uniform and can maintain their state for a long time (long coherence time), but the operation speed is slower compared to superconducting.
Photonic Method: Focused on by PsiQuantum, etc. It uses particles of light (photons). It has the great advantage of not requiring a cryogenic environment and mostly operating at room temperature, while being highly compatible with existing silicon chip manufacturing and fiber-optic communication technologies.
Topological Method: Researched for a long time by Microsoft. It is an ambitious approach trying to create qubits fundamentally robust against environmental noise (resistant to errors) by utilizing the topological properties of special particles called anyons. It is theoretically the strongest, but the physical realization hurdle is considered the highest.

The Path to the Ultimate Goal: “Fault-Tolerant Quantum Computer (FTQC)”

Calculation errors (like bit flips from cosmic rays) also exist in today’s classical computers, but they are perfectly corrected by “error-correcting codes,” so we can use our smartphones without ever noticing an error. To perform practical large-scale calculations on a quantum computer, similar “Quantum Error Correction (QEC)” is essential.

However, because quantum states have the property of “breaking when observed,” there is a fatal dilemma: you cannot directly look inside (observe) to check for errors. To avoid this, theories have been established (like surface codes) that skillfully combine many unstable “physical qubits” to build one stable “logical qubit” that can detect and correct errors. However, it is said that 1,000 to 10,000 physical qubits are needed to create just one logical qubit. Executing Shor’s algorithm using thousands of logical qubits would require a massive system with millions to tens of millions of physical qubits overall.

We are currently in what is called the era of NISQ (Noisy Intermediate-Scale Quantum) devices. These are transitional machines operating with tens to hundreds of qubits without error correction. Experts predict that it will take a long-term research and development period of 10 to several decades to realize the ultimate goal, a fully error-correctable “Fault-Tolerant Quantum Computer (FTQC)”.

Chapter 6: History and Future Prospects of Quantum Computers

Finally, let’s take a bird’s-eye view of how quantum computers were born and where they are heading.

From the Birth of the Theory to the Demonstration of “Quantum Supremacy”

1980s: Physicists Paul Benioff and Richard Feynman proposed the concept of a computer using the principles of quantum mechanics. The phrase “If you want to make a simulation of nature, you’d better make it quantum mechanical” was the starting point.
1994: Peter Shor published a quantum algorithm for prime factorization (Shor’s algorithm). It shocked the world and triggered a massive influx of research funding.
1996: Lov Grover published Grover’s algorithm to speed up data search.
**2019 **: A historical milestone. Google announced that using its 53-qubit superconducting processor “Sycamore,” it completed a random number generation verification calculation in about 200 seconds that would have taken a classical supercomputer 10,000 years (so they claimed). This caused a sensation as the world’s first declaration of demonstrating “Quantum Supremacy” (although IBM and others later improved the classical supercomputer algorithm and argued it could be calculated in a few days, leading to heated debates).
From 2023 onwards: IBM announced the processor “Condor” with over 1,000 qubits. Furthermore, early demonstrations of error correction technology are being reported one after another, such as Harvard University’s success in creating and manipulating “logical qubits.”

Towards the Technology of the Next Generation

A quantum computer is not merely a “next-generation CPU with a faster clock speed.” It is a true paradigm shift in computer science, fundamentally rewriting the concept of computation itself with the rules of quantum mechanics that govern the microscopic world.

We will probably not have a “personal quantum smartphone” that fits in our pockets during our lifetime (and there is no need for one). However, the future is steadily approaching where powerful quantum data centers behind cloud networks like AWS and Azure might one day suddenly discover a cure for an incurable disease, or calculate dream clean energy materials to solve global warming (for example, catalysts to synthesize ammonia from atmospheric nitrogen at room temperature).

We are currently in a dawn comparable to the 1940s ENIAC, which ran on punch cards while the heat from its massive vacuum tubes made the whole room hot. However, top-level researchers and engineers worldwide are pooling their wisdom, and technical breakthroughs are reported daily. We, who can witness the evolution of this new “dawn of computing” in real time, can be said to live in a historically very exciting era.

The door to the quantum world has just been opened. Keep an eye on future developments.

This article aims to explain the basic concepts of quantum computing in an easy-to-understand manner for business professionals and the general public interested in technology. Please note that some rigorous mathematical and physical definitions (such as bra-ket notation and details of complex probability amplitudes) have been partially simplified.

The True Mathematical Structure of the General Number Field Sieve (GNFS)

Sat, 05 Sep 2026 02:26:13 +0900

The True Mathematical Structure of the General Number Field Sieve (GNFS)

The ultimate goal of GNFS is to find $X^2 \equiv Y^2 \pmod N$. To achieve this, mathematicians built a bridge between the “world of real integers” and the “world of algebraic number fields”. That bridge is the “homomorphism”.

Stage 1: The “Homomorphism” Connecting Worlds

1. Polynomial Selection and Root Definition

For a massive composite number $N$, select an integer $m$ and a polynomial $f(x)$ such that $f(m) \equiv 0 \pmod N$. (Example: Expand $N$ in base $m$ and create $f(x)$ from its coefficients. Here, assume $f(x)$ is irreducible (cannot be factored further) over the field of rational numbers $\mathbb{Q}$).

Next, let one of the “complex roots” of the equation $f(x) = 0$ be $\alpha$. Naturally, $f(\alpha) = 0$. $\alpha$ is not an integer but a complex number (an algebraic number) that may involve square roots or imaginary numbers.

2. Construction of Rings and Homomorphisms

Now, prepare two mathematical “rings” (worlds where addition and multiplication are defined).

World A: $\mathbb{Z}[\alpha]$ (Ring of algebraic integers containing $\alpha$) A world of numbers represented in the form $a + b\alpha + c\alpha^2 + \dots$.
World B: $\mathbb{Z}/N\mathbb{Z}$ (Ring of residues modulo $N$) A modulo arithmetic world consisting only of integers from $0$ to $N-1$.

Here, we define a mapping $\phi$ from World A to World B as follows: $$\phi : \mathbb{Z}[\alpha] \to \mathbb{Z}/N\mathbb{Z}$$ $$\phi(\alpha) = m \pmod N$$

This mapping $\phi$ is a magical operation that completely replaces the variable $\alpha$ in World A with the integer $m$ in World B. This $\phi$ possesses an extremely powerful property known as a “Ring Homomorphism”. A homomorphism is the property of “warping to another world without breaking the structure of addition and multiplication”. In other words, the following equations hold:

$\phi(X \times Y) = \phi(X) \times \phi(Y)$
$\phi(X^2) = \phi(X)^2$

What does this mean? If we can create a “square ($\gamma^2$)” of some complex element $\gamma$ in “World A (the world of $\alpha$)”, warping it to “World B (the modulo world)” via $\phi$ will ** perfectly preserve the square form $\phi(\gamma)^2$**.

Stage 2: The Collapse of Prime Factorization and the Birth of the “Ideal”

We want to collect many appropriate elements $(a - b\alpha)$ in World A ($\mathbb{Z}[\alpha]$) and multiply them together to create a “perfect square (square element)”. Normally, we could simply “prime factorize” each collected $(a - b\alpha)$ and combine them (solving via matrices) so that all prime exponents become even, thus creating a square.

However, here a devastating wall of algebra stands in our way. In algebraic number fields like $\mathbb{Z}[\alpha]$, the “uniqueness of prime factorization (any number can be uniquely expressed as a product of primes in exactly one way)”, which we learn in middle school, collapses.

(Example: In a certain algebraic number field, $6 = 2 \times 3$, but at the same time $6 = (1+\sqrt{-5}) \times (1-\sqrt{-5})$, making it impossible to know which are the true primes.)

If prime factorization is not uniquely determined, the puzzle (sieve method) of “counting the number of primes to make them even” becomes fundamentally impossible to execute.

The Salvation by Kummer and Dedekind: The “Ideal”

What saved us from this collapse was the concept of the “Ideal (ideal number)” created by 19th-century mathematicians. By considering not the elements themselves but the “set of multiples (ideals)” generated by those elements, they made prime factorization possible again.

In the ring of integers of an algebraic number field $\mathcal{O}_K$ (a more complete ring containing $\mathbb{Z}[\alpha]$), even if elements cannot be uniquely prime-factorized, it is proven that “an ideal can always be uniquely prime-factorized into a product of ‘prime ideals ($\mathfrak{p}$)’ in exactly one way”.

Therefore, in GNFS, instead of factorizing the element $(a - b\alpha)$ itself, we perform prime ideal factorization on the principal ideal $\langle a - b\alpha \rangle$ that it generates.

Stage 3: Norms and the Two Sieves

So, how do we know which prime ideals the ideal $\langle a - b\alpha \rangle$ factors into? Here, we use a function called the “Norm”. The norm is a function that converts complex elements of an algebraic number field into “ordinary real integers $\mathbb{Z}$”.

The norm of the element $(a - b\alpha)$ can be calculated by a simple polynomial calculation $b^d f(a/b)$ (where $d$ is the degree of $f(x)$).

Due to algebraic theorems, it is known that “if the norm of a certain ideal can be completely factored into small primes (is smooth), then the original ideal can also be completely factored into small prime ideals.”

Thus, GNFS simultaneously calculates the following two values for a massive number of integer pairs $(a, b)$ and collects only the pairs where both values are “smooth numbers”.

Rational Sieve: $a - bm$ (Values in the real world)
Algebraic Sieve: $b^d f(a/b)$ (Norm in the algebraic number field world)

By collecting tens of millions of pairs $(a, b)$ where both are smooth, and solving the prime factorization data of ideals (how many prime ideals are included) as a giant matrix (linear algebra over GF(2)), we find a set $S$ of pairs such that “when multiplied together, the exponents of all prime ideals become even”.

Stage 4: The Two Confronting “Obstructions” and the Ideal Class Group

Through matrix calculations, we found that multiplying all the ideals of $(a - b\alpha)$ belonging to set $S$ results in the square of a certain ideal $I$.

$$\prod_{S} \langle a - b\alpha \rangle = I^2$$

However, it doesn’t end here. The deepest and most difficult mathematical wall in GNFS lies here.

What we ultimately want is not the “square of an ideal”, but the “square of an element ($\gamma^2$)” to plug into the mapping $\phi$. Just because the ideal is squared doesn’t necessarily mean the element itself is squared. There exist two formidable mathematical obstructions here.

Obstruction 1: The Wall of the Ideal Class Group

The ideal $I$ is not always an “ideal generated by a single element (principal ideal)”. It is impossible to extract a concrete element $\gamma$ from an ideal that is not principal.

Here, the concept of the “Ideal Class Group ($Cl_K$)” comes into play. The ideal class group is a group that measures “how many non-principal ideals exist in that algebraic number field (how much the uniqueness of prime factorization is broken)”. Even if $\prod \langle a - b\alpha \rangle$ becomes $I^2$, if $I$ is not the identity element (principal ideal) in the ideal class group, it cannot be pulled back to the square of an element.

Obstruction 2: The Wall of the Unit Group

Suppose we get lucky and $I$ is a principal ideal $\langle \gamma \rangle$. Then, $\prod \langle a - b\alpha \rangle = \langle \gamma^2 \rangle$. You might think, “Great, the element is squared too!”, but that is a huge mistake.

The fact that ideals (sets of multiples) are equal does not mean the elements are completely equal. A deviation by a “Unit (a number whose reciprocal is also an integer, like 1 or -1)” inevitably occurs. In other words, the actual equality of the elements is as follows:

$$\prod_{S} (a - b\alpha) = u \cdot \gamma^2$$

($u$ is an element of the unit group $U_K$)

Unless this unit $u$ itself is the square of something (a square element), the left side can absolutely never become a “perfect square of an element”.

Stage 5: Adleman’s Magic “Quadratic Characters”

The obstruction of the ideal class group and the obstruction of the unit group. How do we overcome these two? Here, a genius method called “Quadratic Characters”, introduced by cryptographer Leonard Adleman (the “A” in RSA) and others, makes its appearance.

To determine “whether a certain element is a perfect square in the algebraic number field”, we use an algebraic number field version of the Legendre symbol (quadratic residue). To the giant matrix from earlier (the puzzle to make the count of prime ideals even), we secretly append dozens of additional conditions (columns) stating that “the quadratic characters for some special prime ideals $\mathfrak{q}$ also all become $1$ (even)”.

When we find a set $S$ that satisfies even these additional conditions through matrix calculations, deep theorems of algebraic number theory guarantee that “both the obstruction of the ideal class group and the obstruction of the unit group will naturally vanish with overwhelming probability.”

With this, we finally obtain the true equation.

$$\prod_{S} (a - b\alpha) = \gamma^2$$

Final Stage: The Convergence of Worlds and the Fall of Encryption

Finally, all the pieces of the puzzle are in place.

[Element in the Algebraic Number Field World (World A)] $\gamma^2 = \prod (a - b\alpha)$ (We find $\gamma$ from this using a square root algorithm)

[Element in the Real World (World of Rational Numbers)] $V^2 = \prod (a - bm)$ (Since this is just ordinary integer multiplication, the square root $V$ can be found normally)

Now, it’s time for the magical bridge we built at the very beginning, the homomorphism $\phi$, to take the stage. We warp the element $\gamma$ from World A to World B (the world modulo $N$) using $\phi$ (the mapping that substitutes $m$ for $\alpha$).

$$Y = \phi(\gamma) \pmod N$$

Meanwhile, we take the $V$ created in the real world directly to the modulo world and call it $X$.

$$X = V \pmod N$$

Due to the homomorphism’s property of “preserving structure”, the square relationship that held in World A is perfectly preserved in World B (the world modulo $N$) as well. Furthermore, because the original pairs $(a, b)$ were correspondingly created in the forms $a - b\alpha$ and $a - bm$, these $X$ and $Y$ collide in the world modulo $N$ to produce the following absolute equation:

$$X^2 \equiv Y^2 \pmod N$$

All that is left is to pray that these $X$ and $Y$ are not trivial solutions ($X \equiv \pm Y$) and calculate: $\gcd(X - Y, N)$

If it is a non-trivial solution, the Euclidean algorithm will race through in 0.001 seconds, and the secret primes $p$ and $q$, which are the heart of RSA encryption, will be printed on the output screen.

This is the complete form of the “General Number Field Sieve (GNFS)”, gathering the essence of modern mathematics.

How to enable KaTeX (LaTeX-style math formulas) in Hugo

Fri, 31 Mar 2023 23:11:26 +0900

What is KaTeX

KaTeX is a JavaScript library for displaying LaTeX-style mathematical formulas in HTML.

Specifically, it can display formulas like the following:

$$f(x) = x^2 + x + 41$$

There seem to be other LaTeX-style math rendering libraries, but KaTeX is known for being simple and fast.

How to introduce it to Hugo

Create a new layouts/partials/math.html in your Hugo folder hierarchy.

Make the contents as follows:

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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.4/dist/katex.min.css" integrity="sha384-vKruj+a13U8yHIkAyGgK1J3ArTLzrFGBbBc0tDp4ad/EyewESeXE/Iv67Aj8gKZ0" crossorigin="anonymous">
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.4/dist/katex.min.js" integrity="sha384-PwRUT/YqbnEjkZO0zZxNqcxACrXe+j766U2amXcgMg5457rve2Y7I6ZJSm2A0mS4" crossorigin="anonymous"></script>
<script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.4/dist/contrib/auto-render.min.js" integrity="sha384-+VBxd3r6XgURycqtZ117nYw44OOcIax56Z4dCRWbxyPt0Koah1uHoK0o4+/RRE05" crossorigin="anonymous"></script>
<script>
document.addEventListener("DOMContentLoaded", function() {
 renderMathInElement(
 document.body,
 {
 delimiters: [
 {left: "$$", right: "$$", display: true},
 {left: "\\[", right: "\\]", display: true},
 {left: "$", right: "$", display: false},
 {left: "\\(", right: "\\)", display: false}
 ]
 });
 });
</script>

Next, add the following code to the existing file layouts/partials/extend_head.html.

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{{ if or .Params.math .Site.Params.math }}
{{ partial "math.html" . }}
{{ end }}

Now you are ready to use KaTeX.

You can enable KaTeX by adding math: true to the front matter of a page.

All you have to do is write formulas in LaTeX style in the body of the page article.

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$$ e^{i \pi} = -1 $$

If you describe it as above, it will be displayed as follows:

$$ e^{i \pi} = -1 $$

References

Favorite Mathematical Formulas

Wed, 05 Oct 2022 13:24:50 +0900

Wilson’s Theorem

When $p$ is a prime number, $(p−1)!≡−1 \pmod p$ holds.

Euler’s Formula

$e^{iπ}+1=0$

Fermat’s Last Theorem

For any integer $n≥3$, there are no positive integer solutions $x, y, z$ to the equation $x^n+y^n=z^n$.

Basel Problem

$1/1^2+1/2^2+1/3^2+1/4^2+...=π^2/6$

Euler’s Prime-Generating Polynomial

$n^2+n+41$ is a prime number for all integers $n$ from $0$ to $39$.

Why do I like math and computers?

Fri, 23 Sep 2022 01:56:52 +0900

I thought about why I like mathematics and computers

Equal to everyone

Regardless of the person, position, or location, it always returns the same output for the same input.

Clear rules

The rules are clearly defined, and there is no absurdity or injustice.

The answers and theories derived are forever correct

There is no worry that an answer once derived will be rewritten later.

Appropriate level of difficulty

There are always unsolved problems, and you can challenge yourself to solve them at any time.

Great Figures in Mathematics

Thu, 22 Sep 2022 23:06:37 +0900

Fermat (P. de Fermat, 1601-1665)

A representative French mathematician of the 17th century. He laid the foundation for analytic geometry, probability theory, and differential calculus. He is also the founder of number theory.

Diophantus (Diophantus, c. 207-c. 291)

A Greek mathematician. Famous for the introduction of symbolic notation and the study of rational solutions. His representative work is “Arithmetica”. The field of finding rational solutions to equations is called Diophantine analysis.

Euler (L. Euler, 1707-1783)

The king of mathematics born in Switzerland. He made achievements in every conceivable area of mathematics. He left behind over 700 papers and 45 books.

Legendre (A. M. Legendre, 1752-1833)

A French mathematician. He left numerous achievements in number theory, elliptic integrals, and the problem of gravitational attraction. The term “number theory” originates from his main work, “Essai sur la théorie des nombres”.

Kummer (E. E. Kummer, 1810-1893)

A German mathematician. He achieved great success by applying the number theory of cyclotomic fields to Fermat’s Last Theorem.

Descartes (R. Descartes, 1596-1650)

A highly original French philosopher. His representative works include “Discourse on the Method”, “Geometry”, and “The World”.

Pascal (B. Pascal, 1623-1662)

A French philosopher known for his research in probability theory and fluids. Later, he joined the Jansenists and engaged in religious activities. “Pensées” is his work from that time.

Bachet (C. G. Bachet, 1581-1638)

A French nobleman who studied number puzzles as a hobby. He published a Latin translation of Diophantus’s “Arithmetica” and included his own research as annotations.

Wallis (J. Wallis, 1616-1703)

An English mathematician. In his main work “Arithmetica Infinitorum”, he gave a mathematical form to the concept of limits, contributing to the foundation of calculus.

Brouncker (W. Brouncker, 1620-1684)

An Irish nobleman. The first President of the Royal Society (British Academy). A mathematics enthusiast who interacted with Wallis and others.

Lagrange (J. L. Lagrange, 1736-1813)

An Italian-born French mathematician. He served as the Director of Mathematics at the Berlin Academy as Euler’s successor. He made major achievements in number theory, analytical mechanics, and more.

Siegel (C. L. Siegel, 1896-1981)

A German number theorist. He made numerous contributions to the analytic theory of numbers.

Fibonacci (L. Fibonacci, c. 1175-c. 1250)

An Italian mathematician. In his main work “Liber Abaci”, he used Hindu-Arabic numerals and worked to popularize them. He also discussed congruent numbers in “Liber Quadratorum”.

Baker (A. Baker, 1939-)

An English number theorist. For his contributions to transcendental number theory, he was awarded the Fields Medal along with Heisuke Hironaka.

Poincaré (J. Poincaré, 1854-1912)

A French mathematician known as the last universal mathematician. He left behind outstanding works in automorphic functions, non-Euclidean geometry, differential equations, celestial mechanics, electrodynamics, and scientific essays.

Mordell (L. J. Mordell, 1888-1972)

An American-born mathematician. His parents were immigrants from Lithuania. He published numerous papers on Diophantine equations, but Mordell’s theorem in the theory of elliptic curves is especially famous. Weil extended this to the case of general algebraic curves, leading to an explosive development in arithmetic algebraic geometry.

Weil (A. Weil, 1906-)

A French mathematician of Jewish descent. In his youth, he took an interest in Fermat’s problem and fundamentally revised algebraic geometry with the aim of applying it to number theory, providing a rigorous foundation. He is also known for his writings on the history of mathematics.

Mersenne (M. Mersenne, 1588-1648)

A French Minim friar. He facilitated correspondence among mathematicians in various places. The numbers known as Mersenne numbers were actually conceived by Fermat.

Roberval (G. P. de Roberval, 1602-1675)

A French mathematician. He was almost the only person at the time who held a professorship as a mathematician, and is known for his research on the cycloid.

Abel (N. H. Abel, 1802-1829)

A Norwegian mathematician. He made outstanding achievements such as proving the insolvability of the general quintic equation, the theory of elliptic functions, and the theory of Abelian integrals, but died at a young age.

Jacobi (K. G. Jacob, 1804-1851)

A German mathematician of Jewish descent. He left major achievements in the theory of elliptic functions, theta functions, differential equations, and the calculus of variations. He introduced Abel’s work to the world. He is also known for stating that the purpose of science is for “the honor of the human spirit,” rejecting utilitarianism.

Gauss (C. F. Gauss, 1777-1855)

The king of mathematics born in Germany. Known for his research in number theory, non-Euclidean geometry, geodesy, hypergeometric series, the theory of surfaces, and geomagnetism. He was a perfectionist who is said to have not published his results unless they were perfected to the point where the scaffolding was invisible. His main work is “Disquisitiones Arithmeticae”.

Lamé (G. Lamé, 1795-1870)

A French mathematician. Contributed to the theory of elasticity, heat conduction, etc.

Liouville (J. Liouville, 1809-1882)

A French mathematician. Contributed to differential equations, transcendental number theory, and complex analysis.

Cauchy (A. L. Cauchy, 1789-1857)

A French mathematician. Highly prolific, with many achievements in theoretical physics and mathematics. In France, he is said to be respected as much as Gauss.

Galois (E. Galois, 1811-1832)

A French mathematician. He elucidated Abel’s theorem that equations of degree 5 or higher generally cannot be solved algebraically by introducing the concept of a group. He died young in a duel, which is now known to have been triggered by a trivial issue involving a woman.

Hensel (K. Hensel, 1861-1941)

A German mathematician. A student of Kronecker. He introduced the theory of p-adic numbers (p-adic analysis) into number theory by applying function-theoretic methods.

Hasse (H. Hasse, 1898-1979)

A German mathematician. A student of Hensel, he contributed to the foundation of p-adic analysis and its applications. His work includes the development of class field theory and research on the L-functions of elliptic curves.

Kronecker (L. Kronecker, 1823-1891)

A German mathematician of Jewish descent. Contributed to the theory of elliptic functions, number theory, etc. Known for advocating the arithmetization of mathematics, but his highly subjective personality led him to unfairly criticize his colleague Weierstrass at the University of Berlin and hinder Cantor’s employment, which has given his life a negative image.

Hilbert (D. Hilbert, 1862-1943)

A German mathematician. Left a massive footprint in every field of mathematics, including invariant theory, algebraic number theory, and the foundations of mathematics. The 23 problems he proposed at the International Congress of Mathematicians in Paris (1900) had a major impact on the development of mathematics.

Riemann (B. Riemann, 1826-1866)

A German mathematician. Made profound contributions to the very foundations of mathematics, such as the concept of Riemann surfaces in complex analysis and Riemannian geometry.

Faltings (G. Faltings, 1954-)

A German mathematician. Resolved the Mordell conjecture in 1983.

Wiles (A. Wiles, 1953-)

An English-born mathematician. A leading figure in this field who made major contributions through his research on Iwasawa’s main conjecture, the Birch and Swinnerton-Dyer conjecture, and famously proving Fermat’s Last Theorem.

References

Fermat’s Last Theorem Solved! From Euler to Wiles’ Proof, Kodansha