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Public key encryption

AP.CSP:
IOC‑2.B (LO)
,
IOC‑2.B.5 (EK)
On the Internet, two computers often want to exchange secure data with each other. When I type my password into the Khan Academy login screen, I want my computer to send that data safely to the Khan Academy servers. I do not want to worry that an attacker might be monitoring my Internet traffic and watching the password go across the wires.
Illustrator of an attacker stealing password data. A laptop is shown with a browser and a password input field. An arrow goes from the laptop to a server. Above the arrow, there's a pouch that contains the text "Be3tP@ssw0rd3ver" and a smiling attacker looking at the text.
Symmetric encryption techniques rely on both the sender and receiver using the same key to encrypt and decrypt the data. How can my computer and the Khan Academy server exchange the key securely? If an attacker can see my password go across the wires, then they can also see an encryption key!
Illustrator of an attacker stealing an encryption key. A laptop is shown with a browser and a password input field. An arrow goes from the laptop to a server. Above the arrow, there's a pouch that contains the text "TOP_SECRET_KEY" and a smiling attacker looking at the text.
Public key encryption to the rescue! It's an asymmetric encryption technique which uses different keys for encryption and decryption, allowing computers over the Internet to securely communicate with each other.
Let's step through the high-level process of public key encryption.

Step 1: Key generation

Each person (or their computer) must generate a pair of keys that identifies them: a private key and a public key.
You can generate a pair below, using the same RSA algorithm that's used by your computer:
Did you notice it takes a few seconds to generate the keys? That's due to the math involved. The keys are generated by multiplying together two incredibly large primes. The algorithm repeatedly generates random large numbers and checks if they're prime, until it finally finds two random large primes. All that checking for primes can take a while, and these keys are only 512 bits long. The current nationally recommended key length is 2048, or even 3072 bits.

Step 2: Key exchange

The sending and receiving computers exchange public keys with each other via a reliable channel, like TCP/IP. The private keys are never exchanged.
Key exchange diagram with a laptop and a server. The laptop screen shows a browser with a password input field. An arrow goes from the server to the laptop and is labeled with "SERVER_PUBLIC_KEY". Under that, an arrow goes from the laptop to the server and is labeled with "CLIENT_PUBLIC_KEY."

Step 3: Encryption

The sending computer encrypts the secret data using the receiving computer's public key and a mathematical operation.
The power of public key encryption is in that mathematical operation. It's a "one-way function", which means it's incredibly difficult for a computer to reverse the operation and discover the original data. Even the public key cannot be used to decrypt the data.
You can try it out below, with the public key you generated above:

Step 4: Sending encrypted data

The sender can now safely transmit the encrypted data over the Internet without worry of onlookers.
Key exchange diagram with a laptop and a server. The laptop screen shows a browser with a password input field. An arrow goes from the server to the laptop and is labeled with string of encrypted text. An attacker looks unhappily at the text.

Step 5: Decryption

Now the receiver can decrypt the message, using their private key. That's the only key that can be used to decrypt the message (in the world!).
Try it out below, with the encrypted message and private key from above:
Once you successfully decrypt the message, try decrypting it with the public key. It won't work; only the private key can decrypt it.

But how is that possible?

It may sound too good to be true; that it's possible to encrypt something with one key that can only then be decrypted by a different key. For a long time, mathematicians weren't sure if it was possible, but fortunately they discovered a way in the 1970s.
The math of the one-way function relies on prime numbers, the difficulty of factoring large primes, and modular arithmetic. If you'd like to dig deeper into the math, check out the Khan Academy tutorials on modern cryptography.
Fortunately, all of us can use and benefit from public key cryptography without needing to understand the complicated math behind it. In fact, we likely use public key cryptography everyday as we use computers and the Internet. Just imagine, what would the world be without it?

🙋🏽🙋🏻‍♀️🙋🏿‍♂️Do you have any questions about this topic? We'd love to answer—just ask in the questions area below!

Want to join the conversation?

  • boggle yellow style avatar for user Joshua
    The decrypt button doesn't show any result. I'm sure I pasted the right information on the fields from the encrypt and decrypt parts.
    Note that I copied all of the text in each key, including the '-----BEGIN __ KEY-----' and '-----END __ KEY-----' parts. I also tried it without those parts, but it still doesn't work.
    (14 votes)
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  • piceratops seed style avatar for user ronanmacg50
    When I encrypt something using my public key, eg "Hi" it seems like it can have many different encrypted forms. How is this possible?
    (3 votes)
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  • hopper cool style avatar for user Alex Ewart
    Is all data sent on the internet due to regulations automatically encrypted?
    When getting other's public keys, generating private keys, and decrypting data, when does all this happen, because I now know that my computer does this but I have no idea when it is all this going on. Is it like when I type in someone's email address? Is the address a public key? If so, is the email address like a way humans can remember the public key like domain names instead of typing IP addresses?
    Does every computer have its own designated keys, or do they change like IP addresses?
    Hope this is not too confusing to answer.
    (3 votes)
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    • blobby green style avatar for user Abhishek Shah
      This is a great question!

      All data sent over the Internet is not encrypted. Only if you use certain protocols like HTTPS will it be encrypted. There is no regulation requiring all data to be encrypted.

      A person's email address is not a public key. Every computer has the ability to create its own keys, but when you get a new computer, it doesn't magically already exist. You have to generate it. Once it's generated, keys don't change. You can always generate a new set of keys though.

      One way I find it easy to think about is the following:

      Think about your home. If someone sends you mail, do they need a key to your home to put it inside? No, they could just slide it under the door or put into your mailbox. When you reach home, you can unlock your home and read the mail.

      This is why as users we don't have to generate keys with our own computers. In some sense, we just send mail to servers by slipping it underneath their doors (via a public key encryption) and they can read it via a private key decryption.

      I hope this helps!
      (3 votes)
  • aqualine ultimate style avatar for user sebastian nielsen
    I don't get how the private and public key looks like a mess. Shouldn't the private and public key just be a number? Then what is all those symbols doing in the private and public key?

    -----BEGIN RSA PRIVATE KEY-----
    MIICWwIBAAKBgH1gajwsAHgJKHD7QEFpzWRSbqA2SxdwpmC/QEdqGZpn4ueGI_REMOVED_SOMEPF4TzF/VAPlJ4IJ6f39oohZU27If3jqStYYY2ctwsQ==
    -----END RSA PRIVATE KEY-----

    P.s. I went through the math but at no point is it mentioned how this strange looking key was derived.
    (3 votes)
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    • blobby green style avatar for user Abhishek Shah
      For convenience, the RSA private key is represented with text. It makes transferring and comparing the keys easier for people. This large text is indeed a large number as there is a one-to-one well-defined encoding between every letter and number. ASCII is one such encoding.
      (2 votes)
  • female robot amelia style avatar for user Aryan
    Hi! I generated my public and private keys, then to tinker around, I encrypted my "hello" message with the private key and tried to decrypt the encrypted message with my public key. The result was "false". Is it the decrypted text that I received or the function returned false?
    (1 vote)
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  • blobby green style avatar for user joe rampulla
    Doesn't seem to work. I generate keys in step 1, take the Public Key from step 1 and use it for step 3, take the text I receive under the Encrypt button and use that, along with the Private Key, in step 5 and nothing happens when I press Decrypt.
    (2 votes)
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  • male robot hal style avatar for user Enn
    Some descriptions of the public key encryption method also mention a "hash function" being applied to the plain text before it is encrypted with the private key ?
    What is the purpose of the hash function ?
    (1 vote)
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    • starky ultimate style avatar for user KLaudano
      A hash function gives an output from a limited range of values when it is given some input. Most hash functions that are used give pseudo-random outputs. (The output is pseudo-random because it is difficult to predict what the output will be when we know the input. However, the hash function will always give the same output when given the same input.) For example, the modulus function is a hash function because x mod n will always give a value between 0 and n.

      Hash functions have a variety of uses, such as providing a way to find elements in a hash table data structure. Hash functions can also be used to verify data integrity. Suppose Bob wants to send a message labelled M to Alice. Bob can apply a hash function to M to get hash(M). Now, Bob sends both M and hash(M) to Alice. Alice will then apply the hash function to the message she received and compares it to hash(M). If the result of the hash function Alice used matches hash(M), then she likely received the correct message. If the results are different, then the message was corrupted or changed in some way during transmission.
      (2 votes)
  • stelly yellow style avatar for user lahwani
    HELLO,
    I think the decrypt button does not work
    (1 vote)
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  • male robot donald style avatar for user bashcove
    Hey Joshua, I don’t know if your still using khan and on this comments page but what's wrong with the atical, i mean, I’m on my iPad but its not letting me decrypt and encrypted my messages so, what...?
    (1 vote)
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  • hopper cool style avatar for user Alex Ewart
    So data sent to your computer is encrypted in a randomly generated code? How does a computer decrypt the sent data? Do you decide your own private key? How do passwords play into this, because at first I thought that your own password was a key and now I am not sure.
    (1 vote)
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    • blobby green style avatar for user Abhishek Shah
      It's great you are thinking about this!

      Passwords are generally used for authentication, not encryption. Authentication seeks to prove who you are. Encryption seeks to hide a message. They are not equivalent.

      As you observed, you can't decrypt the sent data unless you have the right keys. Hence, in protocols like HTTPS where they use public key encryption, the first thing to do is establish the keys. Only then can you decrypt the data.

      I hope this helps and feel free to ask more!
      (1 vote)