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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:
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.
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: and . How such large primes are found: how large primes are found for RSA.
The public key is made of:
  • n = p*q
  • a randomly chosen integer exponent between 1 and e_max = lcm(p -1, q -1), where lcm is the Least common multiple
Given a plaintext message m, the encrypted ciphertext version is:
c = m^e mod n
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 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.
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 is a prime number is .
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.
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 times and send the result over while keeping 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.
Video 1. Elliptic curves by Computerphile (2018) Source. shows the continuous group well, but then fails to explain the discrete part.
ECDH has smaller keys. mentions some interesting downsides:
  • bad curves exist, while in modular, any number seems to work well. TODO why?
  • TODO can't find this mentioned anywher else: Diffie-Hellman key exchange has a proof that there is no algorithm, ECDH doesn't. Which proof?
Ciro Santilli has a hard time understanding why this is possible, e.g. many people use short 4 digit pins, or a short swipe pattern. Why can't this be cracked easily offline?
Can we do better than "wrong password implies random bytes"?
Can the last disk access times be checked via forensic methods?
Generate public private key, test encrypt and test decrypt:
# Create your pubkey.
gpg --gen-key
gpg --armor --output pubkey.gpg --export <myemail>

# Encrypt using someone's pubkey.
gpg --import pubkey2.gpg
echo 'hello world' > hello.txt
gpg --output hello.txt.gpg --encrypt --recipient <other-email> hello.txt

# Double check it is not plaintext in the encrypted message.
grep hello hello.txt.gpg

# Decrypt.
gpg --output hello.decrypt.txt --decrypt --recipient <myemail> hello.txt.gpg
diff -u hello.decrypt.txt hello.txt
This is a way to host a server that actually hide the IP of the server from the client, just like Tor hides the IP of the client from the server. Amazing tecnology!
This is why it enables hosting illegal things like the Silk Road: law enforcement is not able find where the server is hosted, and take it down or identify the owner.
This is where "fun" stuff is likely to be.
An overview of what you can do with a pre-shared key with tradeoffs can be found at: The options are:
In the context of cryptography, authentication means "ensuring that the message you got comes from who you think it did".
Authentication is how we prevent the man-in-the-middle attack.
Authentication is one of the hardest parts of cryptography, because the only truly secure way to do it is by driving to the other party yourself to establish a pre-shared key so you can do message authentication code. Or to share your public key with them if you are satisfied with the safety of post-quantum cryptography.



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