I understand it is possible to convert a Random OLE (ROLE) into an OLE with desired inputs with some extra communication. My understanding is, Alice and Bob mask their desired input with the random inputs from the ROLE and each send a single message as explained in:

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

My question is, can a Vector OLE (VOLE) be converted into multiple OLEs with desired inputs by repeating the same "extra communication" process (as many times as the length of the VOLE)? Or, is the fact that one of Alice's inputs a scalar make it impossible?


1 Answer 1


Short answer: no, there is no obvious way to convert a VOLE to $n$ OLEs.

In more detail, one can view an OLE as an arithmetic generalization of bit OT (bit OT is the case where the field $\mathbb{F} = \mathbb{F}_2)$. Vector OLE can be seen as an arithmetic generalization of string OT, where the receiver still has one choice bit (or scalar in the case of VOLE) but the sender has two bit strings (or vectors in the case of VOLE).

However, note that in both OLE and VOLE, the receiver only has one "choice" scalar, making the VOLE primitive only a generalization for the sender, not the receiver. Since the receiver only has one random choice scalar, it is not obvious how to (information-theoretically) convert a random VOLE correlation into $n$ OLE correlations, even if they might superficially appear similar. The receiver would somehow need to reuse the same random scalar to mask different choice scalars in the "extra communication" phase (i.e., conversion from random OLE to OLE), which would no longer be private since the random mask can only be used once.


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