We consider the reduction loss of security reductions for non-interactive key exchange (NIKE) schemes. Currently, no tightly secure NIKE schemes exist, and in fact Bader et al. (EUROCRYPT 2016) provide a lower bound (of Ω(n~2), where n is the number of parties an adversary interacts with) on the reduction loss for a large class of NIKE schemes. We offer two results: the first NIKE scheme with a reduction loss of n/2 that circumvents the lower bound of Bader et al., but is of course still far from tightly secure. Second, we provide a generalization of Bader et al.'s lower bound to a larger class of NIKE schemes (that also covers our NIKE scheme), with an adapted lower bound of n/2 on the reduction loss. Hence, in that sense, the reduction for our NIKE scheme is optimal.
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