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In a previous question, I described a particular keyed "hash" that mapped a 5-digit input code into a 5-digit output code. It used a 8-bit key which is very insecure - more than 99% of the time, you can infer the key given a single input/output code pair.

I put "hash" in inverted commas as although it has something in common with a hash, it always has an equal input and output length, and it is certainly not secure in anyway.

The input code to output code mapping is not one-to-one. It is possible for an output code to come from several input codes. A given input code only maps to a single output code.

Let's ignore the specific algorithm in the previous question and assume something secure has been developed.

The two most important security properties are:

  1. The attacker should not be able to predict the output code when only given the input code.
  2. The attacker should not be able to derive the key given multiple pairs of input/output codes.

I suspect this means that there are $100000^{100000}$ potential mappings between the input and output codes - essentially limitless. They key length cannot be limited by this constraint.

Does this mean that the key length should obey the normal "long enough to prevent brute forcing" rules that most current encryption does, and should be 128 bits or longer?

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Most of those mappings are crap. Any selected key should give rise to a "good" mapping. And, yes, the key should certainly be large enough to prevent brute-forcing, else someone could try all possible keys and check that they work with a few inputs and outputs. Can you state which security property you are interested in? (what should an attacker not be able to do) –  Thomas May 20 '13 at 12:41
    
I have added the two things an attacker should not possibly do. Whilst a lot of those mappings might be poor, is there a better way of defining that? Is restricting it to one-to-one better? –  Cybergibbons May 20 '13 at 13:16
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Property 2 calls for a sufficiently long key such that brute force is inapplicable. Property 1 requires a nonlinear pseudorandom mapping such that given any number of previous input/output pairs, the attacker cannot predict the output of any new input with probability better than $1/n$ where $n$ is the number of possible outputs. That's how you would define a "poor" and a "good" mapping. Limiting to one-to-one is very bad for property 1, since any new input/output pair simplifies the attacker's job as he can rule out one extra possible output. –  Thomas May 20 '13 at 13:53
    
Thanks. The one-to-one mapping restricting the output set is very clear, I don't know why I didn't think of it. –  Cybergibbons May 20 '13 at 14:40
    
@Thomas: How do you define "crap"? I would say most of those $10000^{10000}$ mappings are fine from a security view, though unfortunately also not really implementable without listing all key-value-pairs (which would look like a 50000-decimal-digit key). –  Paŭlo Ebermann May 20 '13 at 18:17
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up vote 3 down vote accepted

What you want is a pseudo-random function family from the set of 5-digit numbers to itself. As you found out, the total number of such functions is $10000^{10000}$, which is quite a lot.

Contrary to what Thomas said, most of them are not "crap" – a randomly selected function from this space is as secure as we can have. Unfortunately this would mean a key of size 10000 · 5 decimal digits (i.e. around 166096 bits, i.e. 20 KB), which is a bit impractical.

So we need a smaller key, and still a secure selection from the whole space, which just looks like a random selection. That is known as a pseudo-random function family (PRF) ("indexed" by the key). The "looks like random" can be formalized with some indistinguishability criterion, but from that your two criteria follow (each of them allows an easy distinguisher).

The key size should still be big enough to not enable brute-forcing – this nowadays practically means a 128 bit key.

Smaller key sizes might be possible, if you limit the number of queries an adversary can make, but I would not build my security on this – there shouldn't be a problem to store an 128 bit key per device (if one can store secrets there at all).

Of course, the construction of your PRF should be done in a way that doesn't have any inherent weaknesses, like the crap which was used in the example of your previous question.

Using $HMAC(key, input)$ with a secure hash function (like SHA-2), followed by an output transformation which produces five decimal digits, looks like a secure way to build this. As the input has a fixed length and actually fits together with the key in one input block of the hash function (for most crypto hashes), using a simple $H(key || input)$ should work, too.

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My reading had led me to PRFs, but I was not confident that it was the correct term or the right way to think about this. Interestingly, you've touched upon one of my next questions. –  Cybergibbons May 20 '13 at 20:07
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