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For symmetric algorithms (like AES or DES or RC4 -- but not for RSA or ECDSA), a key is a sequence of bits, such that any sequence of bits of the same size is a potential key. So longer keys means more possible keys. Exhaustive search is about trying all possible keys until a match is found. It is an absolute limit to the strength of an algorithm: ...

To answer your first question on key length: DES uses a 56 bit key. A brute-force attack will need a maximum complexity of $2^{56}$ to find the correct key. Now by today's standards this is not much. A complexity of $2^{90}$ or more is considered secure enough. By that standard AES with any key size – 128, 192 or 256 – is strong enough to use. As far as ...

There are a number of distinguishers that it it would be easy to prove are not present in a hash function. For example, I can easily prove that Skein does not have the distinguisher "the 2nd bit in the output is equal to the first bit of the output with probability 1". The proof would be a simple example of a message whose digest does not have this property ...

A longer key length means a greater search space for someone trying to brute force the key. There are $2^{128}$ times more 256-bit keys than 128-bit keys. So, all other things being equal, a brute force search for a 256-bit key could be impractical by a factor of billions in a case where a brute force search for a 128-bit key might be practical. However, ...

Modern block encryption algorithms are so secure that trying to keep the choice of algorithm secret usually results in more harm than good! An encrypted message is garbage if it cannot be successfully decrypted by the recipient. And if you're going to the trouble of encrypting it, that implies it's a very important message. So it's best to make sure there ...