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Public Key Encryption

Mar 19, 2024, 3 min read

The basis behind all sorts of awesome security stuff.

Public key encryption is a form of asymmetric encryption (the key used to encrypt is different from the key used to decrypt) that is the basis for a LOT of modern security. It uses a pair of keys, one public and one private. Either key can be used to encrypt a message that can only be decrypted using the other half. It allows for secure encryption, authentication, and is the basis of RSA, amongst other stuff.

Public Key Encryption depends on Modular Arithmetic, Primitive Roots, and a good One-Way Function.

This uses the ‘Diffie-Hellman’ Key Exchange Algorithm, explained very well in the video in the source. The steps are outlined below.

  1. A and B each have public and private keys.

    A private key = 15
B private key = 13

  2. A & B agree publicly to a Prime Modulus and a Generator

    A: "hey, lets do 3 mod 17"
B: "cool"

  3. A raises their private key to the generator, and sends the result to B

    A does => 3^15 mod 17 ≡ 6
A: "Hey my public key is 6"
B: "Dope"

  4. B does the same

    B does => 3^13 mod 17 ≡ 10
A: "Hey my public key is 12"
B: "Sweet"

  5. A takes the public key from B, raises it to their own private key, and provides the modulus result to B.

    A does => 12^15 mod 17 ≡ 10
A says "I know that our secret is 10"

  6. B does the same procedure with the public key from A and arrives at the same result.

    B does => 6^13 mod 17 ≡ 10
B says "I know that our secret is 10"

At this point, A and B know the shared secret, “10”, but they do not know each other’s private key. What’s more - C, who was listening to all the public information, only has these things to work with: - 3 mod 17 - A sent B “6” - B sent A “12” Using just that info, for sufficiently large numbers, it’s practically impossible to for C to determine secret number “10” now known to A & B.

This has to do with the fact that A and B wound up doing the same Modular Arithmetic, but they “went around the clock” an private number of times.

In reality this is done in conjunction with a pseudorandom generator to encrypt messages between people who’ve never met before.

Source

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