This is a spin-off from Can Big Data attack RSA by just calculating many prime multiplications in advance? [duplicate].
I am somewhat new to cryptography.
Repeating the basics of RSA from How are the primes used to generate RSA keys?:
Textbooks say the one-way function is merely two primes (with some critical constraints) ...
may be a good base of this question, as well as What prime lengths are used for RSA?. Also, Big data and modern crypto systems already asks the same thing as the question here, but in a more abstract way. And the way primes are actually found is also astonishing. Three other links that were recommended in a comment:
- Rainbow tables as a solution to large prime factoring
- How many prime numbers are there (available for RSA encryption)?
- RSA encryption. Breaking 2048 keys with index.
Take "Big Data" and deep neural networks to get over the boundaries of weight and space.
This question brings in Big Data tricks, in contrast to Is it feasible to build an index of prime factors? which asks for an "index" and which does not take into account the chance of stochastically finding out, in a decentralized system, where the prime factors of a chosen outcome are "stored" (meaning, where the possible calculations to get these primes are stored, since storing the numbers is impossible). Quote from the top answer there:
Let's assume for an instant that you could build a large table of all primes. Then... what ? How would you use it ? What would you look up ?
With Big Data tricks, you would not need to scan one single table index in one place, but you could use a Bloom filter and / or a key coordinator (or other Big Data tricks) showing you the right storage or at least guessing the neural network that could calculate what you search for, graphical trees like Merkle trees, whatever. It may still be a huge problem, but take the mere possible links as the physical storage, and not the static particles that build the universe. When taking the links of dynamic neural networks, so that they could store not the outcome, but the mere possible calculation dependent on the input parameters, you lose that "atomic boundary of the universe".
Can Big Data together with deep neural networks attack RSA by affording the vast calculation of prime multiplications in advance?
An example of how such neural networks each could become "experts for a parameter range". This does not mean that it has to be like this in the end.
Think of a neural network that is only for prime numbers that are around the square root of the outcome, with the other networks getting a growing bigger gap between the two primes. Think of taking the unknown primes as the problem of reversing a hash function that might be easier if we train neural networks for a chosen range of the outcome number only, and take the examples that we have from real life to train those networks and split them in the right ranges. These changes should also be possible in the parameters and hyperparameters, not just in the mere input or output of the network.
To my understanding, such Big Data managed dynamic neural networks, at least if the activation function is continuous, could carry up to even infinitely many links with just changes in inputs, hidden layers and gradients, so that the final primes are still just calculated in the end, but you find the neural network that can do it using Big Data.
This link about the question whether the brain has more links than the universe has atoms shows that the number of neurons and the number of links do not say the same thing. And indeed, the brain has about 10^15 links, while the universe has about 10^80 atoms, but it is hopefully clear from this that space and weight lose importance as soon as it comes to links instead of static nodes. (EDITED: This was edited after a comment.)
Taking a continuous activation function (linear activation, ReLU) instead of a discrete "yes / no", and taking the continuous "strength of a link" as a "new link" on its own, then in theory, if we could compute with continuous numbers (which we cannot), just one link would already stand for infinite possible kinds of links - and of course a huge network would have infinite links as well, then.
If weight and space were the problem, anything smaller and lighter than atoms like light or anything physically almost weightless can build unique patterns that work "like a QR code" which then could use neural networks together with Big Data tricks, but that is not what this question is about since atoms are not believed to be the boundary anyway.