The reason why RSA keys are so small is that:
With every doubling of the RSA key length, decryption is 6-7 times times slower.
So this is just another of the security-convenience tradeoffs.
Here’s a graph:
I dug out my copy of Applied Cryptography to answer this concerning symmetric crypto, 256 is plenty and probably will be for a long long time. Schneier explains;
Longer key lengths are better, but only up to a point. AES will have 128-bit, 192-bit, and 256-bit key lengths. This is far longer than needed for the foreseeable future. In fact, we cannot even imagine a world where 256-bit brute force searches are possible. It requires some fundamental breakthroughs in physics and our understanding of the universe.
One of the consequences of the second law of thermodynamics is that a certain amount of energy is necessary to represent information. To record a single bit by changing the state of a system requires an amount of energy no less than kT, where T is the absolute temperature of the system and k is the Boltzman constant. (Stick with me; the physics lesson is almost over.)
Given that k = 1.38 × 10−16 erg/K, and that the ambient temperature of the universe is 3.2 Kelvin, an ideal computer running at 3.2 K would consume 4.4 × 10−16 ergs every time it set or cleared a bit. To run a computer any colder than the cosmic background radiation would require extra energy to run a heat pump.
Now, the annual energy output of our sun is about 1.21 × 1041 ergs. This is enough to power about 2.7 × 1056 single bit changes on our ideal computer; enough state changes to put a 187-bit counter through all its values. If we built a Dyson sphere around the sun and captured all its energy for 32 years, without any loss, we could power a computer to count up to 2192. Of course, it wouldn’t have the energy left over to perform any useful calculations with this counter.
But that’s just one star, and a measly one at that. A typical supernova releases something like 1051 ergs. (About a hundred times as much energy would be released in the form of neutrinos, but let them go for now.) If all of this energy could be channeled into a single orgy of computation, a 219-bit counter could be cycled through all of its states.
These numbers have nothing to do with the technology of the devices; they are the maximums that thermodynamics will allow. And they strongly imply that brute-force attacks against 256-bit keys will be infeasible until computers are built from something other than matter and occupy something other than space.
The boldness is my own addition.
Remark: Note that this example assumes that there is a ‘perfect’ encryption algorithm. If you can exploit weaknesses in the algorithm, the key space might shrink and you’d end up with effectively less bits of your key.
It also assumes that the key generation is perfect – yielding 1 bit of entropy per bit of key. This is often difficult to achieve in a computational setting. An imperfect generation mechanism might yield 170 bits of entropy for a 256 bit key. In this case, if the key generation mechanism is known, the size of the brute-force space is reduced to 170 bits.
Assuming quantum computers are feasible, however, any RSA key will be broken using Shor’s algorithm. (See https://security.stackexchange.com/a/37638/18064)
For one AES is built for three key sizes
128, 192 or 256 bits.
Currently, brute-forcing 128 bits is not even close to feasible. Hypothetically, if an AES Key had 129 bits, it would take twice as long to brute-force a 129 bit key than a 128 bit key. This means larger keys of 192 bits and 256 bits would take much much much longer to attack. It would take so incredibly long to brute-force one of these keys that the sun would stop burning before the key was realized.
That’s a big freaking number. That’s how many possibly keys there are. Assuming the key is random, if you divide that by 2 then you have how many keys it will take on average to brute-force AES-256
In a sense we do have the really big cipher keys you are talking of. The whole point of a symmetric key is to make it unfeasible to brute-force. In the future, if attacking a 256bit key becomes possible then keysizes will surely increase, but that is quite a ways down the road.
The reason RSA keys are much larger than AES keys is because they are two completely different types of encryption. This means a person would not attack a RSA key the same as they would attack an AES Key.
Attacking symmetric keys is easy.
- Start with a bitstring
- Decrypt ciphertext with that bitstring.
- If you can read it, you succeeded.
- If you cannot read it then increment the bitstring
Attacking an RSA key is different…because RSA encryption/decryption works with big semi-prime numbers…the process is mathy. With RSA, you don’t have to try every possible bit string. You try far fewer than
2^2048 bitstrings…but it’s still not possible to bruteforce. This is why RSA and AES keys differ in size.
To sum up everything and answer your question in 1 sentence. We don’t need ridiculously big symmetric keys because we already have ridiculously big symmetric keys. 256 bit encryption sounds wimpy compared to something like a 2048 bit RSA Key, but the algorithms are different and can’t really be compared ‘bit to bit’ like that. In the future if there is a need to longer keys then there will be new algorithms developed to handle larger keys. And if we ever wanted to go bigger on current hardware, it’s simply a time tradeoff. Bigger key means longer decryption time means slower communication. This is especially important for a cipher since your internet browser will establish and then use a symmetric key to send information.