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Time series averaging in dynamic time warping (DTW) spaces has been successfully applied to improve pattern recognition systems. This article proposes and analyzes subgradient methods for the problem of finding a sample mean in DTW spaces. The class of subgradient methods generalizes existing sample mean algorithms such as DTW Barycenter Averaging (DBA). We show that DBA is a majorize-minimize algorithm that converges to necessary conditions of optimality after finitely many iterations. Empirical results show that for increasing sample sizes the proposed stochastic subgradient (SSG) algorithm is more stable and finds better solutions in shorter time than the DBA algorithm on average. Therefore, SSG is useful in online settings and for non-small sample sizes. The theoretical and empirical results open new paths for devising sample mean algorithms: nonsmooth optimization methods and modified variants of pairwise averaging methods.

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This thesis describes mobile proactive secret sharing (MPSS), an extension of proactive secret sharing. Mobile proactive secret sharing is much more flexible than proactive secret sharing in terms of group membership: instead of the group of shareholders being exactly the same from one epoch to the next, we allow the group to change arbitrarily. In addition, we allow for an increase or decrease of the threshold at each epoch. We give the first known efficient protocol for MPSS in the asynchronous network model. We present this protocol as a practical solution to the problem of long-term protection of a secret in a realistic network.

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