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There aren't any 2020, except in the trivial one-time pad case where the key is as large as the message: crypto.stackexchange.com/questions/10815/how-do-we-prove-that-aes-des-etc-are-secure

The only perfect cryptosystem!

The problem is that you need a shared key as large as the message.

Systems like advanced Encryption Standard allow us to encrypt things larger than the key, but the tradeoff is that they could be possibly broken, as don't have any provably secure symmetric-key algorithms as of 2020.

Symmetric-key algorithm is al algorithm implementing symmetric encryption.

2020-so-far yes, Grover's algorithm would only effectively reduce key sizes by half:but there isn't a mathematical proof either.

It allows you to do two things:

One notable application: cryptocurrency, see e.g. how Bitcoin works.

Used for example:

Based on the fact that we don't have a P algorithm for integer factorization as of 2020. But nor proof that one does not exist!

The private key is made of two randomly generated prime numbers: $p$ and $q$. How such large primes are found: how large primes are found for RSA.

The public key is made of:

Given a plaintext message m, the encrypted ciphertext version is:This operation is called modular exponentiation can be calculated efficiently with the Extended Euclidean algorithm.

The inverse operation of finding the private m from the public c, e and $n$ is however believed to be a hard problem without knowing the factors of n.

However, if we know the private p and q, we can solve the problem. As follows.

First we calculate the modular multiplicative inverse. TODO continue.

Bibliography:

Answers suggest hat you basically pick a random large odd number, and add 2 to it until your selected primality test passes.

The prime number theorem tells us that the probability that a number between 1 and $N$ is a prime number is $1/log(N)$.

Therefore, for an N-bit integer, we only have to run the test N times on average to find a prime.

Since say, A 512-bit integer is already humongous and sufficiently large, we would only need to search 512 times on average even for such sizes, and therefore the procedure scales well.

RSA vs Diffie-Hellman key exchange are the dominant public-key cryptography systems as of 2020, so it is natural to ask how they compare:

As its name indicates, Diffie-Hellman key exchange is a key exchange algorithm. TODO verify: this means that in order to transmit a message, both parties must first send data to one another to reach a shared secret key. For RSA on the other hand, you can just take the public key of the other party and send encrypted data to them, the receiver does not need to send you any data at any point.

Based on the fact that we don't have a P algorithm for the discrete logarithm of the cyclic group as of 2020, but we do have an efficient algorithm for modular exponentiation. But nor do we have proof that one does not exist! Living on the edge as usual for public-key cryptography.

ECDH has smaller keys. youtu.be/gAtBM06xwaw?t=634 mentions some interesting downsides: