Definition 1: A key k is called perfect if it is uniformly distributed from the adversary's point of view; a key k is called ε-perfect, if its distribution has an ε trace (statistical) distance to the uniform.
For hash functions, we have following theorem of composition:
Let F be a set of of ε1-AU2 hash functions from M->Z, and let G be a set of ε2-ASU2 hash functions from Z->T. Then H=G*F is an ε-ASU2 hash function family from M->T with ε=ε1+ε2
My Questions is following:
Is there any theorem of composition of ε-perfect keys that is not related to hash functions? To be precise, let us analyze key k1 which is ε1-perfect and key k2 which is ε2-perfect. What is the security (perfectness of the output) of their composition k1 XOR k2? Both keys have the same length.
Simple exor does not work to improve: assume the first three bits are always 0, so these will remain 0 forever, what yu need to look at is extractor theory where somewhat random (perhaps with help of some truly random) is converted to random. This is a rich area of theory on various somewhat random sources, and how to convert them.
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