Public key and private key relationship

Question

I send you data encrypted with pvt1(private) key.

So how do you open this document?

Without my pvt1 how can you open it?

If you can open with your pvt2 key then shouldn't there be a relationship between pvt1 and pvt2 key?

Otherwise how can you open my document?

If you need my pvt1 to open the document then do we meet before to exchange the key?

Answer 1

Notionally, you don't encrypt with a private key. You encrypt with either a public key or a combination of a public key and a private key.

Here's how the most common scheme works:

Encrypting with a public key:
1. The recipient generates a public/private key pair. These two keys are related.
2. The recipient sends you their public key and you confirm that it belongs to them.
3. You encrypt something with the recipient's public key.
4. This can only be decrypted with the recipient's private key which only the recipient has.
5. The recipient decrypts the message with their private key.

The advantage of this system is that no setup is required between the sender and the recipient specifically. The recipient can make their public key available to all and that allows anyone to send them an encrypted message that only they can read.

The public key and private key are related in some special way. It's complicated math, but here's an oversimplified (but still tricky) explanation of how it's typically done:

You have some operation that behaves like multiplication except it is irreversible. The private key is generated randomly. There is some well known constant G. The public key is the private key "multiplied" by G. I'll call the recipient's secret key Rs and their public key Rp. So Rp = Rs * G. (Here * represents an operation that is like multiplication but is irreversible.)

To encrypt a message, you choose a new, random public/private key pair just to use for this message. I'll call the public key Ep and the private key Es. So Ep = Es * G.

To send a message, you encrypt it with Es * Rp and send it along with Ep. To decrypt it, the recipient decrypts it with Rs * Ep.

Why does this work? Because Es * Rp = Rs * Ep. Why?
Es * Rp = Rs * Ep
Es * Rs * G = Rs * Es * G
This is true because this multiplication-like operation is commutative.

Why can't an attacker decrypt it? An attacker has:
Ep (because we sent it)
Rp (because it's public)

And an attacker needs either Es * Rp or Rs * Ep. That is, he needs Es * Rs * G and he only has Es * G and Rs * G. Oops, he's stuck since he can't divide.