The Easy Math Behind Public Key Cryptography

The unique model of this story appeared in Quanta Magazine.

For 1000’s of years, when you needed to ship a secret message, there was just one approach to do it. You’ll obfuscate the message through the use of a particular rule, recognized solely to you and your target market. This rule acted like the important thing to a lock. If you happen to had the important thing, you could possibly decipher the message; in any other case, you’ll have to decide the lock. Some locks are so efficient that they’ll by no means be chosen, even with infinite time and sources. However even these methods endure from the identical Achilles heel that afflicts all encryption methods of this kind: how do you get that key into the appropriate palms whereas holding it away from the unsuitable ones?

The counterintuitive answer, referred to as public key cryptographyshouldn’t be about holding a key secret however slightly about making it extensively out there. The trick is to additionally use a second key that you just by no means share with anybody, even the individual you might be speaking with. Solely through the use of this mix of two keys, one public and one personal, can somebody each scramble and decrypt a message.

To grasp how this works, it is best to think about “keys” not as objects that match right into a lock, however as two complementary elements in invisible ink. The primary ingredient makes messages disappear, the second makes them reappear. If a spy named Boris needs to ship a secret message to his counterpart Natasha, he writes a message then makes use of the primary ingredient to make it invisible on the web page. (It is simple for him: Natasha has revealed a easy and well-known system for making the ink disappear.) When Natasha receives the paper within the mail, she applies the second ingredient which causes Boris’s message to reappear.

On this scheme, anybody could make messages invisible, however solely Natasha could make them seen once more. And since she by no means shares the system for the second ingredient with anybody, not even Boris, she will ensure that the message was not deciphered alongside the best way. When Boris needs to obtain secret messages, he merely adopts the identical course of: he publishes a simple recipe for making the messages disappear (which Natasha or anybody else can use), whereas holding one other one for himself that may make them disappear. reappears.

In public-key cryptography, the “public” and “personal” keys work precisely like the primary and second elements of this particular invisible ink: one encrypts messages, the opposite decrypts them. However as an alternative of utilizing chemical substances, public key cryptography makes use of mathematical puzzles referred to as hatch functions. These features are simple to calculate in a single path and intensely troublesome to reverse. However additionally they comprise “trapdoors”, info which, if recognized, makes features trivially simple to calculate in both path.

A standard trapdoor perform is to multiply two massive prime numbers, an operation that’s simple to carry out. However reversing it, that’s, beginning with the product and discovering every prime issue, is computationally impractical. To create a public key, begin with two massive prime numbers. These are your trapdoors. Multiply the 2 numbers collectively, then do some extra calculations. mathematical operations. This public key can now encrypt messages. To decrypt them, you have to the corresponding personal key, which comprises the prime components, the required trapdoors. With these numbers it’s simple to decipher the message. Hold these two major components secret and the message will stay secret.