Public-key cryptography

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.

The algorithm is completely analogous to Diffie-Hellman key exchange in that you efficiently raise a number to a power $n$ times and send the result over while keeping $n$ as private key.

The only difference is that a different group is used: instead of using the cyclic group, we use the elliptic curve group of an elliptic curve over a finite field.

Variant of Diffie-Hellman key exchange based on elliptic curve cryptography.

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