Let $F$ be a PRF.
The $XOR\text{-hash}$ is defined as:
XOR-hash(k, m): //m=(a₀, a₁, ..., aᵥ)
t = 0ⁿ
for each aᵢ ∈ m:
t = t ⊕ F(k, aᵢ)
return t
Suppose an attacker $\mathcal{A}$ can send 3-uples $(m_0, m_1, \delta)$.
A function $f$ will be difference-unpredictable if $\mathcal{A}$ cannot guess a 3-uple such that $f(k,m_0) - f(k,m_1) = \delta \;$ with more than negligible chance.
How can I show that $XOR\text{-hash}$ is difference-unpredictable?
I think the argument relies on the fact that $\oplus$ keep all the bits pseudo-random, so that the difference $d$ will look like a subtraction of two random results, hence the probability of some bit of $d$ match some bit of $\delta$ will be $\small\sim$$0.5$ and the probability of $d = \delta$ would be negligible.
