# Converting a random OLE (oblivious linear function evaluation) to an OLE

It is known that by adding an extra round of communication, it is possible to convert a random OT (where the choice bits and the sender input are random) to a standard OT, see https://crypto.stackexchange.com/a/84206/48273 for details. OLE (oblivious linear function evaluation) is a generalization of OT. Specifically, for a random OLE, Alice gets output $$(x', a')$$ and Bob gets output $$(y', b')$$, where each element are in a finite field satisfying $$x'y' = a' + b'$$.

My question is: if Alice and Bob are given the tuples above, how do they perform OLE using their own input $$x$$ from Alice and $$y$$ from Bob so that $$xy = a+b$$ for some $$a$$ and $$b$$ using one round of communication?

Given the random OLE $$(x',a'), (y',b')$$:
• Alice sends $$u = x+x'$$ and Bob sends $$v = y+y'$$.
• Alice outputs $$\alpha = a'-x'v$$ and Bob outputs $$\beta = b' + uy$$.
$$\alpha + \beta = a'+b' + uy - x'v = x'y' + (x+x')y - x'(y+y') = xy$$,
and (perfect) security follows from the fact that $$x',y'$$ are uniformly random and perfectly mask $$x,y$$ (over, say, a finite field or a finite ring).