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An open subset$D$of$R$^{$d$},$d≧2$, is called Poissonian iff every bounded harmonic function on the set is a Poisson integral of a bounded function on its boundary. We show that the intersection of two Poissonian open sets is itself Poissonian and give a sufficient condition for the union of two Poissonian open sets to be Poissonian. Some necessary and sufficient conditions for an open set to be Poissonian are also given. In particular, we give a necessary and sufficient condition for a Greenian$D$to be Poissonian in terms of its Martin boundary.

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Research on key mismatch attacks against lattice-based KEMs is an important part of the cryptographic assessment of the ongoing NIST standardization of post-quantum cryptography. There have been a number of these attacks to date. However, a unified method to evaluate these KEMs’ resilience under key mismatch attacks is still missing. Since the key index of efficiency is the number of queries needed to successfully mount such an attack, in this paper, we propose and develop a systematic approach to find lower bounds on the minimum average number of queries needed for such attacks. Our basic idea is to transform the problem of finding the lower bound of queries into finding an optimal binary recovery tree (BRT), where the computations of the lower bounds become essentially the computations of a certain Shannon entropy. The optimal BRT approach also enables us to understand why, for some lattice-based NIST candidate KEMs, there is a big gap between the theoretical bounds and bounds observed in practical attacks, in terms of the number of queries needed. This further leads us to propose a generic improvement method for these existing attacks, which are confirmed by our experiments. Moreover, our proposed method could be directly used to improve the side-channel attacks against CCA-secure NIST candidate KEMs.