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The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain. 

Abstract Physical unclonable functions (PUFs) are emerging as an alternative to information security by providing an advanced level of cryptographic keys with non‐replicable characteristics, yet the cryptographic keys of conventional PUFs are not reconfigurable from the ones assigned at the manufacturing stage and the overall authentication process slows down as the number of entities in the dataset or the length of cryptographic key increases. Herein, a supersaturated solution‐based PUF (S‐PUF) is presented that utilizes stochastic crystallization of a supersaturated sodium acetate solution to allow a time‐efficient, hierarchical authentication process together with on‐demand rewritability of cryptographic keys. By controlling the orientation and the average grain size of the sodium acetate crystals via a spatiotemporally programmed temperature profile, the S‐PUF now includes two global parameters, that is, angle of rotation and divergence of the diffracted beam, in addition to the speckle pattern to produce multilevel cryptographic keys, and these parameters function as prefixes for the classification of each entity for a fast authentication process. At the same time, the reversible phase change of sodium acetate enables repeated reconfiguration of the cryptographic key, which is expected to offer new possibilities for a next‐generation, recyclable anti‐counterfeiting platform.