This paper proposes and proves security of a QKD protocol which uses
two-universal hashing instead of random sampling to estimate the number of bit
flip and phase flip errors. For this protocol, the difference between
asymptotic and finite key rate decreases with the number $n$ of qubits as
$cn^{-1}$, where $c$ depends on the security parameter. For comparison, the
same difference decreases no faster than $c’n^{-1/3}$ for an optimized protocol
that uses random sampling and has the same asymptotic rate, where $c’$ depends
on the security parameter and the error rate.

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