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It think the common analogy is a padlock and a key. One party gets the padlock and the other one the key. Now with the padlock you can just lock the box but not open it. And with the key you can just unlock and open the box. That's assymetric. You hand out padlocks (your public key) to everyone, but keep the key (your private key).
The maths behind that is a bit difficult to explain but not that difficult. I think it's about one-way functions that are easy to calculate in one direction and impossible to solve the other way around. There are several ways to do it. Like the old approaches with prime numbers. It's easy to multiply two prime numbers. But given an arbitrary large number, it's difficult to tell which prime numbers it consists of.
https://www.baeldung.com/cs/prime-numbers-cryptography
So using the formula in that guide, you get a numerical value for O. But surely someone else could follow the same process and also get the same answer? Unless the primes change each time? But then how would the sender and receiver know the way in which the values change?
I'm not sure I get your question... Sure other people can also follow the same process and encrypt stuff to you. They can also do the calculations with your private key and arrive at the same result, sure. But the calculation involves your private key. Your secret. If that's known to someone, they can do the calculations. In the example you need to keep the "53" a secret and give the "221" to other people. Everyone with the "53" can decrypt. Everyone with the "221" can encrypt. It's just with the "221" you can't decrypt. That part of the calculation needs people to put in the other number.
I explained it poorly - what I mean to say is, two people trying to send the message 'Hello' for example both using the same public key would get the same output. So if you had a simple message like that, someone could work out by checking every word in the dictionary what your message was by checking if the output matched.
But I guess it's a bit of a moot point - it's unlikely that an encrypted message would ever be so simple. It could just as easily be much longer, and therefore basically impossible to guess the plaintext.
Ah, that is a really good question. These things happen. People have entire harddisks filled with "rainbow tables" which do these kind of attacks against hash-functions which are supposed to be one-way functions. This way they have terabytes worth of pre-computed hashes for the most common passwords and can immediately tell if one of those passwords is in a database leak.
For this it needs additional measures. Passwords are augmented with additional random data so people can't pre-compute the hashes. So it wouldn't be just 'Hello', but 'Hello' plus an additional (random) "salt" that gets fed into the one-way function so it can't be brute forced.
PGP for example uses both symmetric cryptography and asymmetric cryptography. The actual message is encrypted with symmetric encryption and the key to that is encrypted with asymmetric encryption. Unfortunately it's been a while since I last read a book on cryptography. I think they did that because symmetric cryptography is way faster. But things like that could also prevent such attacks. If you use the asymmetric encryption just for decrypting the other randomly chosen key which encrypts the actual message... There is no way to guess that. You'd have to check not just a small dictionary or know the plaintext, but check every random number in existence.
It's not always obvious to the layman what kinds of attacks are possible with the crypto algorithms. They definitely need to protect against such scenarios or they're worthless for that kind of use. There are "known plaintext attacks". Usually people don't want anyone even able to prove that you sent a certain message. And an algorithm also isn't good if you can learn something about the secret key if you have access to both a ciphertext and plaintext (the other variables in the equation). I think this was part of how they cracked the supposedly secure enigma machines of the Nazis. A proper modern cryptography algorithm protects against any of that. Sometimes by combining several things.
Edit: Sure and I forgot padding which @mumblerfish said. And appending data additionally helps if you don't want someone to know the exact length of a message. Or an algorithm sometimes can't encrypt arbitrary amounts of plaintext but does it in fixed block lengths...
Ah thanks for the useful links! Those articles are all quite fascinating. In the plaintext attacks article, I love the tactic mentioned here:
Both cryptography and that part of history are fascinating topics. I can also recommend watching "The Imitation Game" with Benedict Cumberbatch starring as Alan Turing... I mean it's just a movie and skips lots of the interesting stuff and details. YMMV.
It's the beginning of computers. And I think especially that time has some interesting stories, discoveries/inventions and personas. There is also the history and role of women in computing which I think is something more people should know about and it's related to that. After that we needed secrecy in the cold war. I think public key cryptography hasn't been around until the 1970s. There had been export regulations on cryptography until after I was born. And modern encryption algorithms like AES are from the 1990s. Nowadays everyone and their grandma relies on the availability of secure communications.
I think I spent some nights jumping from Wikipedia article to Wikipedia article and reading all of that.
No, it can be very important. As I answered in another comment, its called padding https://en.m.wikipedia.org/wiki/Padding_(cryptography). And to see that it is imortant say you encrypt your easy to remember password in an encrypted file. Now if your attack was possible, having your public key, you could just generate the passwords and encrypt them to figure out your password. Much easier than trying to find your key. Using forms of padding, that does not work.
This post on Stack Overflow does a good job of explaining the issue.