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Public-key cryptography / digital signatures / one-way functions

Public-key cryptography / digital signatures / one-way functions

This glossary provides explanations of various keywords to help you understand the concepts necessary for successful data utilization and digital transformation (DX).
This time, we will explain public-key cryptography, a crucial technology that underpins the very safety and security of the internet.

What are public-key cryptography and digital signatures?

Public-key cryptography is an encryption technique in which the password used to encrypt data (public key) and the password used to decrypt the encrypted data (private key) are different. It is widely used as the foundation for technologies that enable secure and reliable data transfer over the internet.

An electronic signature is a technology that makes it possible to prove that a specific declaration of intent has been made on digital data, similar to a signature or seal on a paper document. It is achieved by using the private and public keys of public-key cryptography in reverse. It is widely used as a means of realizing legal declarations of intent on digital data and is one of the technologies that make blockchain possible.

A one-way function is a key technical element in public-key cryptography. It's a process where the forward operation is practically feasible, but the corresponding reverse operation is computationally intensive and practically impossible to perform. Specifically, the difficulty of prime factorization is sometimes utilized in one such example.

⇒Electronic signature | Glossary

What is "public-key cryptography"?

Public-key cryptography is a technology that underpins the security and safety of the current internet (or IT, or modern society itself). It is one of the most important technologies in the modern world, without which society could not function.

However, public-key cryptography has many aspects that are not intuitively easy to understand. Therefore, I would like to explain what it is, why it became necessary, and how it is technically impressive.

What comes to mind when you hear the phrase "encrypting data"?

"Public-key cryptography" is a cryptographic technique with certain characteristics. In other words, what we're going to talk about is cryptography. So, what comes to mind when you hear "encrypting data"? For example, you might imagine something like "being able to keep data secret using a password, and thereby making data handling secure."

  • encryption
    • Set a password for the "original data to be encrypted" (plaintext) and then encrypt it.
    • Those who do not know the password will not be able to recover their data.
    • Encryption protects the confidentiality of data.
  • decrypt
    • The encrypted data (ciphertext) is decrypted using a password.
    • By reverting to the original data (plain text), the data can be used again.

To reiterate, "How is cryptography used?", it would be something like this.

This article provides a detailed explanation of these "traditional encryption methods" (symmetric-key cryptography).
⇒ Symmetric-key cryptography / DES / AES (Advanced Encryption Standard) | Glossary

The usage is almost the same, but "public-key cryptography" uses "two types of passwords".

The "public-key cryptography" we'll be discussing today is largely the same in terms of its purpose as described above. However, a major difference is that it uses two different types of "passwords."​

  • Encryption (using public-key cryptography)
    • The original data to be encrypted (plaintext) is encrypted by setting a "public key" (a type of password).
    • Encryption protects the confidentiality of data.
  • Decryption (using public-key cryptography)
    • The encrypted data (ciphertext) is decrypted using a private key (a type of password).
    • By reverting to the original data (plain text), the data can be used again.

In short, public-key cryptography is a type of encryption where the password used for encryption and the password used to decrypt the original data are separate. These passwords also have the following names:

  • Public Key: The password used when encrypting data.
  • Secret Key: The password used for the plaintext (original data).

The above is a "minimum explanation" of "how public-key cryptography works." However, you might still not fully understand it and wonder, "Why bother making it complicated by using two types of keys (passwords) when one would suffice?" or "What is something like this used for?"

However, I will now explain how this technology, which may even seem to have an unclear purpose, is actually an extremely important technology that underpins the safety and security of the internet itself, or IT itself.

"Symmetric-key cryptography" cannot provide the security and peace of mind necessary for online shopping.

Nowadays, the use of IT has become widespread throughout society. It is used not only in business but also in daily life. You are reading this article via the internet, but one of the problems with using the internet is that "the security of data during transmission is not guaranteed."

The internet is an open communication network in which "anyone in the world can participate as part of the communication infrastructure," and that is why it has developed so rapidly to cover the entire world. However, on the other hand, it is unknown who manages the communication along the way or where it passes through, and there is a risk that data may be intercepted or tampered with along the way.

"Encryption" to enable the transmission of important data over the internet.

The internet was originally intended for academic use, connecting universities around the world, so the possibility of data being intercepted during transmission wasn't a major issue. However, if you want to use it for business, that's a problem.

This is where encryption technology comes in. If you encrypt the data before sending it, even if the data is intercepted during transmission, its contents will not be understood. However, a major limitation is the big problem of "how to deliver the password used for encryption to the recipient of the data."

  • The internet means that data can be intercepted during transmission.
  • Encrypting data before transmission eliminates the problem of data being intercepted during transmission, making it suitable for business use.
  • Therefore, it is necessary to securely deliver the "password" used for encryption.

Since encryption technology is necessary precisely because data transmission over the internet is insecure, passwords cannot be sent over the internet. Therefore, sharing passwords through "some other secure means" is desirable.

In other words, the situation where "data can be securely delivered from person A to person B through encryption" is actually based on the premise that "there is another means by which person A can securely share the password with person B," and it becomes clear that encryption does not create any new security or safety, but merely "reuses existing security and safety."

How can I safely and securely send data to someone I've just met online?

This limitation becomes a major problem, for example, when you want to "create a website and implement online sales."

In order to accept orders and process payments via credit card, it is necessary to obtain confidential information such as names and addresses, as well as sensitive information such as credit card details. However, it is unrealistic to expect to "securely share passwords with customers who visit your online shop in advance."

In other words, a system that "encrypts and sends with a password" would only allow secure communication with people you already know outside of the internet, essentially making it an "exclusive service" to everyone. That wouldn't be viable for business.

"Public-key cryptography" solves the problem.

Public-key cryptography is the solution to this problem. As already explained, public-key cryptography is an encryption technology in which the key (password) used to encrypt data and the key used to decrypt the encrypted data are different.

By using public-key cryptography as described below, it becomes possible to achieve encrypted data transfer even without the premise of sharing passwords in advance.

  • Generate a key pair
    • The password (public key) used when encrypting
    • The password (private key) used to decrypt encrypted data.
  • The password used for encryption will be made publicly available on the internet.
  • Those who wish to send me data:
    • Encrypt the data using the "password used for encryption" that is publicly available online.
    • The password to decrypt the encrypted data is different and has not been made public, so it is not possible to recover the original data.
    • Data can be transmitted while maintaining its safety and security.
  • The received "encrypted data" can be decrypted using a "password used to decrypt the data" that has not been made public.

Furthermore, because of this usage, the password used for encryption is called the "public key," and the password used to decrypt encrypted data is called the "secret key."

However, the realization of "public-key cryptography" faces fundamental difficulties.

It's a very well-thought-out system, but the explanation so far has only described how "it would be wonderful if such a system existed." Without the "technical mechanism" to actually implement such a system, it's nothing more than a pipe dream.

In reality, it's not uncommon for excellent requests from businesses to be technically impossible to implement. Public-key cryptography also has its share of fundamental difficulties (technically impractical aspects) that make it seem to the average person that "it's probably not feasible."

You can solve it by calculating in reverse, right?

When using public-key cryptography for encryption, the encryption process is performed using a "publicly available key (password)" and a "publicly available algorithm," meaning that "there is absolutely nothing secret about the data."

  • Symmetric-key cryptography (classical cryptography): AES encryption, etc.
    • Encryption algorithm: Publicly available
    • Password used for encryption: confidential information
  • Public-key cryptography
    • Encryption algorithm: Publicly available
    • Password used for encryption: Publicly available

With symmetric-key cryptography, the assumption is that the password is unknown. However, with public-key cryptography, the password is not secret. Therefore, it is impossible to create a secret element, such as "what specific calculations were performed during the encryption process."

  • common key cryptography
    • Original data + password → calculation process → calculation process → ... → calculation process → encrypted data
    • The process itself isn't a secret, but the parameters (passwords) used in the process are.
  • Public-key cryptography
    • Original data + public key → calculation → calculation → ... → calculation → encrypted data
    • The process itself is not a secret, nor are the parameters (public keys) used in the process secret, so everything about what was done is completely transparent.

In other words, public-key cryptography, by its very nature, inevitably reveals "exactly what calculations were performed to encrypt the data." If all the operations performed are known, then it becomes possible to "reverse all the operations" to return to the original data. If the original operation was "multiply by three," then "divide by three," and if it was "add seven," then "subtract seven." By performing these operations in reverse, it's obvious that you can return to the original data, right?

  • An example of "just do it in reverse."
    • Original data + public key → Operation A (multiply by 3) → Operation B (add 7) → ... → Operation C (swap bits according to the rule) → Encrypted data
    • Original data ← Inverse operation A (divide by 3) ← Inverse operation B (subtract 7) ← ... ← Inverse operation C (perform the bit swap operation in reverse) ← Encrypted data + public key

I understand that it would be wonderful if this could be achieved, but the fact that it can be deciphered by working backwards makes it seem inherently difficult to realize. At least for an ordinary person, it seems like a tough situation.

"One-way functions" that implement "public-key cryptography"

However, methods have been devised to address this situation, and public-key cryptography has been put into practical use.

So how did they solve it? Well, they knew what the inverse operation to be performed was known, but they inserted a process where the amount of computation required for the calculation was such that it was practically impossible to perform the inverse operation within a practical timeframe. In other words, even though they knew what the process should be, they made it practically impossible to perform the calculation, thus solving the problem.

*For situations where a calculation method exists in theory but is practically impossible to implement due to the excessive computational complexity, please also see this article.
⇒ Algorithms / Computational complexity theory / P / NP | Glossary

  • The key concept behind the realization of "public-key cryptography"
    • Original data → ... → Operation X (can be calculated) → ... → Encrypted data
    • Original data ← ... ← Inverse operation X (too computationally intensive to be practically calculated) ← ... ← Encrypted data

In other words, the process of "moving from A to B" can be processed with a practical amount of computation, but the reverse process of "moving from B back to A" requires too much computation to be practically calculated, and therefore the code cannot be deciphered.

Processes with these properties are called "one-way functions," and this is, so to speak, the key idea that makes public-key cryptography, which has now become "a technology that supports the very safety and security of modern society," possible.

Concrete examples of "unidirectional functions"

A clear and easy-to-understand example of a "one-way function" is "prime factorization."

While multiplying prime numbers by prime numbers can be done with a practical amount of computation, the reverse operation, which involves prime factorization and returning to the original prime number, requires a massive amount of calculation, making it practically impossible to find the answer when dealing with very large numbers.

  • Forward calculations (which can realistically be done by hand)
    • Multiplication of prime numbers (actually prime numbers):
      104381 × 103687 = 10822952747
  • Calculation in the reverse direction (theoretically possible, but practically difficult to calculate)
    • Factorize 10822952747:
      That's a bit...

Multiplying two six-digit prime numbers is easily done by hand. However, the reverse operation, prime factorization of an eleven-digit number, becomes impossibly difficult to do by hand.

In public-key cryptography, which is widely used today, prime factorization is often used as a "one-way function" (other methods such as the "discrete logarithm problem" and the "elliptic curve discrete logarithm problem" are also sometimes used).

  • Public-key cryptography
    • Original data → ... → Multiplication of large prime numbers (can be calculated) → ... → Encrypted data
    • Original data ← ... ← Prime factorization of large composite numbers (too computationally intensive to be practically feasible) ← ... ← Encrypted data

Public-key cryptography is an extremely important technology that underpins security and safety in IT, and the key to its implementation lies in the "one-way function." Specifically, "prime factorization" is often used.

When you learned about prime numbers and prime factorization in school math class, you might have wondered, "What's the point of this?" But in reality, prime factorization is such an important element that modern society couldn't function without it.

Public-key cryptography is also a means of realizing "digital signatures."

What we've discussed so far is impressive enough, but public-key cryptography is also a means of achieving yet another extremely important aspect of "securing IT safety and security."

It is also used to implement "electronic signatures," a function that is absolutely essential for conducting business with digital data, similar to how "signatures" and "stamps" fulfilled the role of "signatures" and "stamps" in paper documents.

What if we reversed the "public key" and the "private key"?

Public-key cryptography is a technology where the password used for encryption and the password used to decrypt the encrypted data are different. In previous discussions, we used the encryption password as the public key. But what would happen if we reversed the public key?

  • encryption
    • The "original data to be encrypted" (plaintext) is encrypted by setting a "private key" on it.
    • Since it's a private key, only I can encrypt it.
  • decrypt
    • The encrypted data (ciphertext) is decrypted using the public key.​
    • Anyone can decrypt encrypted data, but no one can encrypt it.

Now, conversely, "only I can encrypt it, and anyone can decrypt it back to the original data," which is "a nonsensical situation that's like something out of a comedy sketch" as a means of keeping information secret.

However, by "rethinking" this situation in the following way, we were able to realize a very important use case: an "electronic seal" for electronic data, rather than encryption.

  • Electronic signature (encryption)
    • It's not a means of concealing data, but rather a way of "stamping that data," in other words, a "means of expressing one's intentions."
    • The original data to be digitally signed (plaintext) is encrypted using a private key. Since I am the only one who possesses the private key, only I can perform this process (digital signing/encryption).
  • Verification (decryption) of the digital signature
    • Anyone can use a "public key" to decrypt "digitally signed/encrypted data" (ciphertext), and anyone can "confirm and verify" that it "matches the original data."
    • Someone who does not possess the private key cannot create such "signed (encrypted) data."
    • Therefore, it serves as proof that "the owner of the private key performed an action on that data that only the owner could perform" (i.e., that person expressed their intention).

How to use electronic signatures

For example, let's say you have a contract created in Word. Unlike paper documents, electronic data can be copied and its contents can be altered. If a contract is created as electronic data, and its contents can be altered later, it becomes impossible to know what agreements were made.

Therefore, even when a contract is created electronically, it is sometimes printed on paper, signed by hand, stamped with a seal, and then mailed. The fundamental issue is that "digital data can be tampered with," so even if you create a function to "embed an image that looks like a seal on Word," the seal image can be altered, so it cannot replace a seal.

However, using public-key cryptography, it becomes possible to make "declarations of intent" using only electronic data, similar to how a seal is used.

  • Create contracts as electronic data using programs like Word.
  • By encrypting the "contract data" with "my private key" (the encryption key), a "digitally signed contract" is created. Since this process cannot be performed by anyone else, it becomes objectively verifiable that the key owner has "expressed their intent."
  • Anyone other than me can use "my public key" (the key to decrypt the encrypted data) to convert the "digitally signed contract" back into "contract data."
  • As a result, others can confirm that the "contract data" has undergone "processing using a private key that only I can access," and that an expression of intent has been made using a private key.

*In actual use of electronic signatures, the contract data itself is not encrypted with a private key (because the amount of data is too large to process). Instead, a hash function is used to obtain a hash value (digest value) from the contract data, and this value is then electronically signed (encrypted), thereby achieving essentially the same result.

Electronic signatures are widely used as a means of constructing legal expressions of intent on digital data, such as expressing agreement to contracts and declaring ownership of data, including copyright. They are also a very important fundamental technology that broadly supports modern society.

For example, in the realization of blockchain, which makes up cryptocurrencies, digital signatures using public-key cryptography are a crucial technological element. For instance, a Bitcoin "address" (similar to an account number in a bank account) is actually the "public key for the digital signature." Ownership of a specific item on the blockchain is declared by the fact that "a digital signature has been made using the private key corresponding to that public key (address)."

Public-key cryptography as an example of how "useful applications" and "technology" are successfully combined.

What's "quite rare" about public-key cryptography is not only that it achieves advanced technology, but also that its applications (on a business side) have been well considered.

In the real world, "technology and its application (business) often don't go together well."

Japan prides itself on being a technology-driven nation, and many companies throughout Japan undoubtedly prioritize technological capabilities and are actively pursuing technological development. However, a common problem in technological development is that "technology" and "application (business)" often fail to align effectively.

Even when research and development of technology is successful, it's often met with comments like, "I don't understand what it's useful for." This is inherently the nature of technological development, so it's difficult to simply tell people to focus only on profitable research and development.

On the other hand, just because something is deeply rooted in a need doesn't necessarily mean it will work. We can come up with all sorts of innovative ideas, such as "If there was a robot that could automatically take in and fold laundry," but many of these are simply not feasible from a technical standpoint, and forcing such requests can easily lead to technical problems.

Considering the common challenges associated with such technological development, public-key cryptography seems like a rare example of a truly remarkable combination of useful applications and the achievement of technically challenging feats.

The key is to effectively combine what is "possible" (technical capabilities) with what "will result if implemented" (needs).

That's why I think people say you need to understand both technology and business. However, I think it's realistically difficult for me or anyone else to do what Steve Jobs did.

If a comprehensive, high-level approach is difficult, then perhaps what we can do is to recognize and combine the things we "can do" (technical capabilities) and the things that "would change if implemented" (needs) that we often think, "it would be great if we could do this," while simultaneously having those thoughts in our daily work.

Alternatively, every company may have many things they "can do" and many things they "would like to change (requests)" but are not utilizing them because they "haven't realized how to combine them."

"Connecting" technology plays a vital role as a means of effectively combining and utilizing existing elements.

In this discussion, if you're thinking, "If electronic signature technology exists, I'd definitely like to use it," then what's needed in actual implementation is to "find a good way to combine it."

In reality, starting to use electronic signatures involves either using electronic signature services offered as various cloud services or utilizing the functions included in packaged software. However, simply being able to create electronic signatures is not enough to achieve results.

Electronic signatures are applied to "some kind of data," but that data is likely scattered across "other systems and clouds" throughout the company, in various locations and formats.

Furthermore, since results can be achieved by finding ways to effectively utilize these features in one's own business activities, it is necessary to integrate "data" and "electronic signature functionality" into "business," and it will be necessary to experiment and refine the use cases that produce results through trial and error in the field.

Please utilize "connecting" technology

There are ways to efficiently develop these various "integration processes" using only a GUI. These are "connecting" technologies such as "DataSpider" and "HULFT Square," also known as "EAI," "ETL," and "iPaaS." By utilizing these, you can smoothly and efficiently integrate old and new systems (see the end of the article for details).

The threat looming over public-key cryptography, which underpins "IT safety and security" (quantum computing)

Another point to keep in mind is that a significant portion of the "security and safety of IT systems in the world" actually depends on public-key cryptography. Or, the security of the "one-way function" (often the difficulty of prime factorization) that makes public-key cryptography possible is a broad prerequisite for achieving security and safety related to IT.

As IT becomes increasingly integrated into society and business, it's becoming more and more important to consider what would happen "if IT becomes unusable." For example, would a company survive if all its data were lost due to a system failure? Or, could a cloud service experience a massive outage? These are questions that "every organization should consider."

Public-key cryptography is a fundamental technology that underpins so many things. It is the basis for secure and reliable communication on the internet, and it is also the foundation for legal actions such as digital declarations of intent and digital declarations of ownership. For example, just as a major outage at AWS would have a tremendous impact on the world, something happening to public-key cryptography would have a tremendous impact on the world.

And what is currently a real concern is the impact of "quantum computers moving towards practical application." If quantum computers become fully practical, "prime factorization will no longer be a difficult calculation," meaning that the security of widely used public-key cryptography may not be maintained.

*I plan to write about this topic soon. I hope you will take a look at it as well.
⇒Quantum-resistant cryptography | Glossary

Related keywords (for further understanding)

  • common key cryptography

  • Electronic Signature

  • Algorithms / Computational complexity theory / P / NP

  • Post-Quantum Cryptography (PQC)

  • EAI
    • It is a concept of "connecting" systems by data integration, and is a means of freely connecting various data and systems. It is a concept that has been used since long before the cloud era as a way to effectively utilize IT.
  • ETL
    • In the recent trend of actively working on data utilization, the majority of the work is not the data analysis itself, but rather the collection and preprocessing of data scattered in various places, from on-premise to cloud.
  • iPaaS
    • A cloud service that "connects" various clouds with external systems and data simply by operating on a GUI is called iPaaS.

Are you interested in "iPaaS" and "connecting" technologies?

Try out our products that allow you to freely connect various data and systems, from on-premise IT systems to cloud services, and make successful use of IT.

The ultimate "connecting" tool: data integration software "DataSpider" and data integration platform "HULFT Square"

"DataSpider," data integration tool developed and sold by our company, is a "connecting" tool with a long history of success. "HULFT Square," a data integration platform, is a "connecting" cloud service developed using DataSpider technology.

Another feature is that development can be done using only the GUI (no code) without writing code like in regular programming, so business staff who have a good understanding of their company's business can take the initiative to use it.

Try out DataSpider/ HULFT Square 's "connecting" technology:

There are many simple collaboration tools on the market, but this tool can be used with just a GUI, is easy enough for even non-programmers to use, and has "high development productivity" and "full-fledged performance that can serve as the foundation for business (professional use)."

It can smoothly solve the problem of "connecting disparate systems and data" that hinders successful IT utilization. We regularly hold free trial versions and hands-on sessions where you can try it out for free, so we hope you will give it a try.


Why not try a PoC to see if "HULFT Square" can transform your business?

Why not try verifying how "connecting" can be utilized in your business, the feasibility of solving problems using data integration, and the benefits that can be obtained?

  • I want to automate data integration with SaaS, but I want to confirm the feasibility of doing so.
  • We want to move forward with data utilization, but we have issues with system integration
  • I want to consider data integration platform to achieve DX.

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