The simple math behind public key cryptography

The original version from this story appeared Quanta Magazine.

For thousands of years, if you wanted to send a secret message, there was basically one way to do it. You would type the message using a special rule known only to you and your intended audience. This rule acted as a key to the lock , you can decrypt the message. otherwise, you must pick the lock. Some locks are so efficient that they can never be picked, even with infinite time and resources.But even these schemes suffer from the same Achilles’ heel that plagues all such encryption systems;

A controversial solution known as public key cryptographyrelies not on keeping the master secret, but on making it widely available. The trick is to also use a second key that you never share with anyone, not even the person you’re communicating with. Only using this combination of two keys one public and one private, someone can both decrypt and decrypt the message.

To understand how this works, it’s easier to think of “keys” not as objects that fit into a lock, but as two complementary components of invisible ink A spy named Boris wants to send a secret message to his colleague Natasha, he writes a message and then uses the first component to make it invisible on the page. Natasha has published an easy and popular formula for disappearing ink.) When Natasha receives the paper in the mail, she applies a second ingredient that makes Boris’s message reappear.

With this scheme, anyone can make messages invisible, but only Natasha can make them visible again. And since she never shares the formula for the second component with anyone, not even Boris, she can be sure that the message was not deciphered on the way wants to receive secret messages, he simply accepts the same procedure; he publishes an easy recipe to make the messages disappear (that Natasha or anyone else can use) while keeping another one just for himself that makes them reappear.

In public key cryptography, the “public” and “private” keys work as the first and second components in this special invisible ink; one encrypts messages and the other decrypts But instead of using chemicals, public key encryption uses mathematical puzzles called trap functions. These functions are easy to compute in one direction and extremely difficult to invert, but they also contain “traps,” information that, if known, makes the functions easy to compute in both directions.

A common trapdoor function involves multiplying two large primes. But its inverse, that is, starting with the product and finding each prime factor, is computationally infeasible. To make a public key, these are your traps : Multiply the two numbers together, then do some additions mathematical operations. This public key can now encrypt messages. To decrypt them, you need the corresponding key factors, the necessary traps. With those numbers, it is easy to decrypt the message, and the message will remain secret.