  Two party differential privacy allows two parties who do not trust each
other, to come together and perform a joint analysis on their data whilst
maintaining individual-level privacy. We show that any efficient,
computationally differentially private protocol that has black-box access to
key agreement (and nothing stronger), is also an efficient,
information-theoretically differentially private protocol. In other words, the
existence of efficient key agreement protocols is insufficient for efficient,
computationally differentially private protocols. In doing so, we make progress
in answering an open question posed by Vadhan about the minimal computational
assumption needed for computational differential privacy.
  Combined with the information-theoretic lower bound due to McGregor, Mironov,
Pitassi, Reingold, Talwar, and Vadhan in [FOCS'10], we show that there is no
fully black-box reduction from efficient, computationally differentially
private protocols for computing the Hamming distance (or equivalently inner
product over the integers) on $n$ bits, with additive error lower than
$O\left(\frac{\sqrt{n}}{e^{\epsilon}\log(n)}\right)$, to key agreement.
  This complements the result by Haitner, Mazor, Silbak, and Tsfadia in
[STOC'22], which showed that computing the Hamming distance implies key
agreement. We conclude that key agreement is \emph{strictly} weaker than
computational differential privacy for computing the inner product, thereby
answering their open question on whether key agreement is sufficient.
