first pair of people who can be mistaken for each other by the system – once you have somewhat over a thousand people enrolled. It may well, however, be OK to use it to verify a claimed identity (though many other things can go wrong; see the chapter on Biometrics in Part 2 for a discussion).
There are some applications where collision-search attacks aren't a problem, such as in challenge-response protocols where an attacker has to find the answer to the challenge just issued, and where you can prevent challenges repeating. In identify-friend-or-foe (IFF) systems, for example, common equipment has a response length of 48 to 80 bits. You can't afford much more than that, as it costs radar accuracy.
But there are other applications in which collisions are unacceptable. When we design digital signature systems, we typically pass the message
For this reason, hash functions used with digital signature schemes have
To sum up: if you need a cryptographic hash function to be collision resistant, then you'd better choose a function with an output of at least 256 bits, such as SHA-2 or SHA-3. However if you only need to be sure that nobody will find a second preimage for an existing, externally given hash, then you can perhaps make do with less.
5.3.2 Random generators – stream ciphers
The second basic cryptographic primitive is the random generator, also known as a keystream generator or stream cipher. This is also a random function, but it's the reverse of the hash function in that it has a short input and a long output. If we had a good pseudorandom function whose input and output were long enough, we could turn it into a hash function by throwing away all but a few hundred bits of the output, and turn it into a stream cipher by padding all but a few hundred bits of the input with a constant and using the output as a keystream.
It can be used to protect the confidentiality of our backup data as follows: we go to the keystream generator, enter a key, get a long file of random bits, and exclusive-or it with our plaintext data to get ciphertext, which we then send to our backup service in the cloud. (This is also called an additive stream cipher as exclusive-or is addition modulo 2.) We can think of the elf generating a random tape of the required length each time he is presented with a new key, giving it to us and keeping a copy on his scroll for reference in case he's given the same input key again. If we need to recover the data, we go back to the generator, enter the same key, get the same keystream, and exclusive-or it with our ciphertext to get our plaintext back again. Other people with access to the keystream generator won't be able to generate the same keystream unless they know the key. Note that this would not give us any guarantee of file integrity; as we saw in the discussion of the one-time pad, adding a keystream to plaintext can protect confidentiality, but it can't detect modification of the file. For that, we might make a hash of the file and keep that somewhere safe. It may be easier to protect the hash from modification than the whole file.
One-time pad systems are a close fit for our theoretical model, except in that they are used to secure communications across space rather than time: the two communicating parties have shared a copy of a keystream in advance. Vernam's original telegraph cipher machine used punched paper tape; Marks describes how SOE agents’ silken keys were manufactured in Oxford by retired ladies shuffling counters; we'll discuss modern hardware random number generators in the chapter on Physical Security.
A real problem with keystream generators is to prevent the same keystream being used more than once, whether to encrypt more than one backup tape or to encrypt more than one message sent on a communications channel. During World War II, the amount of Russian diplomatic traffic exceeded the quantity of one-time tape they had distributed in advance to their embassies, so it was reused. But if
To avoid this, the normal engineering practice is to have not just a key but also a seed (also known as an initialisation vector or IV) so we start the keystream at a different place each time. The seed
5.3.3 Random permutations – block ciphers
The third type of primitive, and the most important in modern cryptography, is the block cipher, which we model as a random permutation. Here, the function is invertible, and the input plaintext and the output ciphertext are of a fixed size. With Playfair, both input and output are two characters; with DES, they're both bit strings of 64 bits. Whatever the number of symbols and the underlying alphabet, encryption acts on a block of fixed length. (So if you want to encrypt a shorter input, you have to pad it as with the final ‘z’ in our Playfair example.)
We
