ADD

Suppose that value of \(X, Y\) are shared (using random value) by Alice and Bob.
Alice and Bob compute “secure ADD” independently, and get the shares of the \(Z\).
The value of \(Z = X + Y\) is shared after the “secure ADD” protocol.

\[
\begin{align}
& & \mbox{ Alice } & &\mbox{ Bob } & & \mbox{ Constraint/Result} \\
\mbox{Input } & & X^A, Y^A & & X^B, Y^B & & X^A+X^B = X, Y^A+Y^B = Y\\
\mbox{secure ADD} && Z^A=X^A+Y^A && Z^B=X^B+Y^B && Z^A+Z^B=A+B\\
\end{align}
\]

MULT

Suppose that value of \(X, Y\) are shared (using random value) by Alice and Bob.

Preprocessing

\[
\begin{align}
&& \mbox{ Alice } &&\mbox{ Bob } && \mbox{ Constraint} \\
\mbox{create} && && && \Delta_X \Delta_Y=\Delta_Z (\Delta_X, \Delta_Y \mbox{random})\\
\mbox{store } && \Delta_X^A && \Delta_X^B && \Delta_X^A+\Delta_X^B = \Delta_X \\
\mbox{store } && \Delta_Y^A && \Delta_Y^B && \Delta_Y^A+\Delta_Y^B = \Delta_Y \\
\mbox{store } && \Delta_Z^A && \Delta_Z^B && \Delta_Z^A+\Delta_Z^B = \Delta_Z \\
\end{align}
\]

Secure MULT

\begin{align}
\mbox{Input } && X^A, Y^A && X^B, Y^B && X^A+X^B = X, Y^A+Y^B = Y\\
\mbox{Exchange} && X^A-\Delta_X^A && X^B-\Delta_X^B && \\
\mbox{Exchange} && Y^A-\Delta_Y^A && Y^B-\Delta_Y^B && \\
\mbox{Reconstruct} && X-\Delta_X && X-\Delta_X && X-\Delta_X= (X^A-\Delta_X^A)+(X^B-\Delta_X^B)\\
\mbox{Reconstruct} && Y-\Delta_Y && Y-\Delta_Y && Y-\Delta_Y= (Y^A-\Delta_Y^A)+(Y^B-\Delta_Y^B)\\
\mbox{Output } & & Z^A & & Z^B & & Z^A+Z^B=Z \\
\end{align}
\]
The “secure MULT” calculation is:
\[
Z_A = (x-\Delta_X)(Y-\Delta_Y) + (X-\Delta_X)\Delta_Y^A + (Y-\Delta_Y)\Delta_X^A + \Delta_Z^A \\
Z_B = (X-\Delta_X)\Delta_Y^B + (Y-\Delta_Y)\Delta_X^B + \Delta_Z^B
\]