What is Public Key Cryptography?

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Cryptography Historically

It's helpful to take a look back at the study of cryptography over the years to understand where and when digital signatures came into play.

Historically, up until the 1970s, cryptography was the study of encrypting messages so that they could not be decrypted even if intercepted. Cryptography was used for passing important secrets, especially within the military.

The sender would take their message and pass it through a function to create an encrypted output. This could be something as simple as moving each character in a string down one position in the alphabet. For instance "abc" becomes "bcd". This function is not particularly difficult to crack, especially once you know its secret! 🧐

As cryptography advanced over the years, more complex functions were introduced to hide messages better. One important leap forward was the idea of a secret key. 🔑

If two parties can meet prior to their exchange of messages they both can come to an agreement on a particular key. This key plus a function (like the alphabet shift mentioned above) could be used together to create a more secure encryption. 🔑

👆🏼 Having keys on both sides of the message is considered symmetric-key cryptography.

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As the study of cryptography advanced, methods continued to become more complex and creative. A good place to start to learn more about these methods is this Wikipedia Article.

Personal Computing

As mentioned above, the state-of-the-art cryptography had been increasingly complex versions of symmetric-key cryptography. The objective of the game was to never let your adversary have your key. And, of course, in military situations, countries were doing everything in their power to break your code! 🔐

With the advent of personal computing on the horizon, some cryptographers started to think outside the box. People would want to be able to communicate with each other securely without having to worry about eavesdroppers. Of course, they could meet somewhere in person and exchange keys any time people wanted to talk securely, but this seemed a bit old-fashioned. This got cryptographers thinking: How could two parties communicate securely without having met beforehand to exchange keys?

This was a tough problem! Was it even possible? 🤔

In 1976 Whitfield Diffie proposed an idea. What if there was a public key? 💡

Many established cryptographers dismissed this idea out of hand. After all, the whole purpose of an encryption key was to keep it private!

As it turns out, a public key introduces some extremely important cryptographic properties! Take a minute to walk through this thought experiment with me conceptually:

Start Thought Experiment 🧠

Let's say there is a private key that can decrypt a message from a public key and vice-versa. Each key is the only key that can decrypt a message encrypted by the other key.

Now imagine Bob has declared a public key far and wide as the key that identifies him. Bob will keep a private key that corresponds to his public key. When he uses his private key to encrypt a message, he can share it publicly to be decrypted using his public key. Upon decrypting this message, we can say beyond the shadow of a doubt that only Bob could have written this message. The only key that could have encrypted the message is the corresponding private key which only Bob has access to. In practice, this would create an unforgeable digital signature for Bob.

On the flip side, what if a message was encrypted using Bob's public key? Of course, anyone can do this since Bob's public key is available to everyone. The benefit comes in that only Bob can decrypt the message. In this way, a friend of Bob's can write a message that can only be read by Bob. They could send it through any network, regardless of its security so long as it reaches Bob. They could rest assured that nobody would be able to decrypt the message except for Bob.

End Thought Experiment 🧠

👆🏼 This was the insight that struck Whitfield Diffie in 1976. The only problem was, he didn't have any practical way to make this happen. He had a concept, but he did not have a mathematical function with these properties! Diffie would work with both Martin Hellman and Ralph Merkle in search of such a system.

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It's kind of mind-blowing to think that this invention started with a concept before finding a mathematical function that could satisfy these properties.

As opposed to the encryption techniques mentioned in the above section, public key cryptography is considered asymmetric encryption in that only one party has access to the private key.

RSA and ECDSA

Today, both RSA and ECDSA are two popularly used algorithms for public key cryptography.

The RSA algorithm is based on the idea that it's very easy to find the product of two prime numbers, yet extremely difficult to factor out those two prime numbers if you have the product. You can see this example on Wikipedia for a good rundown.

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The Mathematics behind these algorithms can be quite difficult to wrap your brain around! How difficult the RSA algorithm is to decrypt is still an unsolved mystery in Computer Science. It's assumed it can only be decrypted in exponential time (relative to the size of the input), which essentially boils down to a brute-force attack of randomly guessing at the key. See P Versus NP Problem.

The ECDSA algorithm uses elliptic curves. It can provide the same level of security as other public key algorithms with smaller key sizes, which is the reason it's become quite popular. It is the Digital Signing Algorithm used by Bitcoin, specifically the secp256k1 curve.

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