which hash functions are orthogonal to each other?

Question

I'm interested in multi-level data integrity checking and correcting. Where multiple error correcting codes are being used (they can be 2 of the same type of codes). I'm under the impression that a system using 2 codes would achieve maximum effectiveness if the 2 hash codes being used were orthogonal to each other.

Is there a list of which codes are orthogonal to what? Or do you need to use the same hashing function but with different parameters or usage?

I expect that the first level ecc will be a reed-solomon code, though I do not actually have control over this first function, hence I cannot use a single code with improved capabilities.

Note that I'm not concerned with encryption security.

Edit: This is not a duplicate of

When are hash functions orthogonal to each other? due to it essentially asking what the definition of orthogonal hash functions are. I want examples of which hash functions that are orthogonal.

@BlackVegetable no, that is a different question, which essentially asks what the definition of orthogonal hash functions are. I want examples of which hash functions are orthogonal — BeowulfNode42, Feb 13 '14 at 01:57
@BlackVegetable If you're going to delete your comment about [When are hash functions orthogonal to each other?](http://stackoverflow.com/questions/21709464/when-are-hash-functions-orthogonal-to-each-other) can you also delete the duplicate mark on the question...? — BeowulfNode42, Feb 13 '14 at 08:23
I'm not sure how to do that. I'll look on meta and see if it is possible. Sorry about all of this. — BlackVegetable, Feb 13 '14 at 15:12
It is not possible to undo a flag. I apologize for the inconvenience. In return, I'll try to answer your question! (See: http://meta.stackexchange.com/a/206859/188241) — BlackVegetable, Feb 13 '14 at 15:14
I've had a question on askubuntu SE site that had the duplicate mark removed somehow... Maybe it was voted off by other people... — BeowulfNode42, Feb 13 '14 at 19:20

score 1 · Answer 1 · edited May 23 '17 at 12:17

I'm not certain it is even possible to enumerate all orthogonal hash functions. However, you only asked for some examples, so I will endeavour to provide some as well as some intuition as to what properties seem to lead to orthogonal hash functions.

From a related question, these two functions are orthogonal to each other:

Domain: Reals --> Codomain: Reals
f(x) = x + 1
g(x) = x + 2

This is a pretty obvious case. It is easier to determine orthogonality if the hash functions are (both) perfect hash functions such as these are. Please note that the term "perfect" is meant in the mathematical sense, not in the sense that these should ever be used as hash functions.

It is a more or less trivial case for perfect hash functions to satisfy orthogonality requirements. Whenever the functions are injective they are perfect hash functions and are thus orthogonal. Similar examples:

Domain: Integers --> Codomain: Integers
f(x) = 2x
g(x) = 3x

In the previous case, this is an injective function but not bijective as there is exactly one element in the codomain mapped to by each element in the domain, but there are many elements in the codomain that are not mapped to at all. These are still adequate for both perfect hashing and orthogonality. (Note that if the Domain/Codomain were Reals, this would be a bijection.)

Functions that are not injective are more tricky to analyze. However, it is always the case that if one function is injective and the other is not, they are not orthogonal:

Domain: Reals --> Codomain: Reals
f(x) = e^x // Injective -- every x produces a unique value
g(x) = x^2 // Not injective -- every number other than 0 can be produced by two different x's

So one trick is thus to know that one function is injective and the other is not. But what if neither is injective? I do not presently know of an algorithm for the general case that will determine this other than brute force.

Domain: Naturals --> Codomain: Naturals
j(x) = ceil(sqrt(x))
k(x) = ceil(x / 2)

Neither function is injective, in this case because of the presence of two obvious non-injective functions: ceil and abs combined with a restricted domain. (In practice most hash functions will not have a domain more permissive than integers.) Testing out values will show that j will have non-unique results when k will not and vice versa:

j(1) = ceil(sqrt(1)) = ceil(1) = 1
j(2) = ceil(sqrt(2)) = ceil(~1.41) = 2
k(1) = ceil(x / 2) = ceil(0.5) = 1
k(2) = ceil(x / 2) = ceil(1) = 1

But what about these functions?

Domain: Integers --> Codomain: Reals
m(x) = cos(x^3) % 117
n(x) = ceil(e^x)

In these cases, neither of the functions are injective (due to the modulus and the ceil) but when do they have a collision? More importantly, for what tuples of values of x do they both have a collision? These questions are hard to answer. I would suspect they are not orthogonal, but without a specific counterexample, I'm not sure I could prove that.

These are not the only hash functions you could encounter, of course. So the trick to determining orthogonality is first to see if they are both injective. If so, they are orthogonal. Second, see if exactly one is injective. If so, they are not orthogonal. Third, see if you can see the pieces of the function that are causing them to not be injective, see if you can determine its period or special cases (such as x=0) and try to come up with counter-examples. Fourth, visit math-stack-exchange and hope someone can tell you where they break orthogonality, or prove that they don't.

which hash functions are orthogonal to each other?

1 Answers1