  Oblivious linear evaluation is a generalization of oblivious transfer,
whereby two distrustful parties obliviously compute a linear function, f (x) =
ax + b, i.e., each one provides their inputs that remain unknown to the other,
in order to compute the output f (x) that only one of them receives. From both
a structural and a security point of view, oblivious linear evaluation is
fundamental for arithmetic-based secure multi-party computation protocols. In
the classical case, oblivious linear evaluation protocols can be generated
using oblivious transfer, and their quantum counterparts can, in principle, be
constructed as straightforward extensions using quantum oblivious transfer.
Here, we present the first, to the best of our knowledge, quantum protocol for
oblivious linear evaluation that, furthermore, does not rely on quantum
oblivious transfer. We start by presenting a semi-honest protocol and then
extend it to the dishonest setting employing a commit-and-open strategy. Our
protocol uses high-dimensional quantum states to obliviously compute f (x) on
Galois Fields of prime and prime-power dimension. These constructions utilize
the existence of a complete set of mutually unbiased bases in prime-power
dimension Hilbert spaces and their linear behaviour upon the Heisenberg-Weyl
operators. We also generalize our protocol to achieve vector oblivious linear
evaluation, where several instances of oblivious linear evaluation are
generated, thus making the protocol more efficient. We prove the protocols to
have static security in the framework of quantum universal composability.
