  Consider a PPT two-party protocol $\pi=(A,B)$ in which the parties get no
private inputs and obtain outputs $O^A,O^B\in \{0,1\}$, and let $V^A$ and $V^B$
denote the parties' individual views. Protocol $\pi$ has $\alpha$-agreement if
$Pr[O^A=O^B]=1/2+\alpha$. The leakage of $\pi$ is the amount of information a
party obtains about the event $\{O^A=O^B\}$; that is, the leakage $\epsilon$ is
the maximum, over $P\in\{A,B\}$, of the distance between $V^P|OA=OB$ and
$V^P|OA\neq OB$. Typically, this distance is measured in statistical distance,
or, in the computational setting, in computational indistinguishability. For
this choice, Wullschleger [TCC 09] showed that if $\alpha>>\epsilon$ then the
protocol can be transformed into an OT protocol.
  We consider measuring the protocol leakage by the log-ratio distance (which
was popularized by its use in the differential privacy framework). The
log-ratio distance between X,Y over domain \Omega is the minimal $\epsilon>0$
for which, for every $v\in\Omega$, $log(Pr[X=v]/Pr[Y=v])\in
[-\epsilon,\epsilon]$. In the computational setting, we use computational
indistinguishability from having log-ratio distance $\epsilon$. We show that a
protocol with (noticeable) accuracy $\alpha\in\Omega(\epsilon^2)$ can be
transformed into an OT protocol (note that this allows $\epsilon>>\alpha$). We
complete the picture, in this respect, showing that a protocol with $\alpha\in
o(\epsilon^2)$ does not necessarily imply OT. Our results hold for both the
information theoretic and the computational settings, and can be viewed as a
"fine grained" approach to "weak OT amplification".
  We then use the above result to fully characterize the complexity of
differentially private two-party computation for the XOR function, answering
the open question put by Goyal, Khurana, Mironov, Pandey, and Sahai [ICALP 16]
and Haitner, Nissim, Omri, Shaltiel, and Silbak [FOCS 18].
