Let$X$(-ϱ$B$^{$m$}×$C$^{$n$}be a compact set over the unit sphere ϱ$B$^{$m$}such that for each$z$∈ϱ$B$^{$m$}the fiber$X$_{$z$}={ω∈$C$^{$n$};($z, ω$)∈$X$} is the closure of a completely circled pseudoconvex domain in$C$^{$n$}. The polynomial hull $$\hat X$$ of$X$is described in terms of the Perron-Bremermann function for the homogeneous defining function of$X$. Moreover, for each point ($z$_{0},$w$_{0})∈Int $$\hat X$$ there exists a smooth up to the boundary analytic disc$F$:Δ→$B$^{$m$}×$C$^{$n$}with the boundary in$X$such that$F$(0)=($z$_{0},$w$_{0}).

We study to what extent rearrangements preserve the integrability properties of higher order derivatives. It is well known that the second order derivatives of the rearrangement of a smooth function are not necessarily in$L$^{1}. We obtain a substitute for this fact. This is done by showing that the total curvature for the graph of the rearrangement of a function is bounded by the total curvature for the graph of the function itself.

Two post-quantum password-based authenticated key exchange (PAKE) protocols were proposed at CTRSA 2017. Following this work, we give much more efficient and portable C++ implementation of these two protocols. We also choose more compact parameters providing 200-bit security. Compared with original implementation, we achieve 21.5x and 18.5x speedup for RLWE-PAK and RLWE-PPK respectively. Compare with quantum-vulnerable J-PAKE protocol, we achieve nearly 8x speedup. We also integrate RLWE-PPK into TLS to construct a post-quantum TLS ciphersuite. This allows simpler key management, mutual authentication and resistant to phishing attack. Benchmark shows that our ciphersuite is indeed practical

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