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The log-optimal strategy in gambling and asset allocation problem attempts to maximize the expectation of the logarithmic rate of return but not the gross wealth itself. This strategy has been shown to have long termsuperiority overother strategies in theory. Another remarkable virtue is it allows updating the real-time market information quickly which could increase of the logarithmic rate of return. We will prove these properties ourselves in a simple way. We further consider how to do asset allocation by applying the log-optimal strategy in practice. The distribution function of the returns is usually unknown and need to be estimated from the history in a real world. The estimation error will affect the asset allocation directly, but such effect mayresult in a "butterfly effect" which could bring an investment disaster. Therefore, it is an important issue to answer whether there is logoptimal strategy resisted "butterfly effect". We develop a new log-optimal strategy based on the allocation utility which is a quadratic approximation to the expectation of the logarithmic rate of return. We show that the upper bound of the allocation utility deviation can be controlled by the l_1-norm of portfolio and l_∞-norm of the bias of variance matrix estimator. This turns out that our proposed strategy is robust. We apply our strategy for NYSE data. The results showed that our strategy has high performance in return.

We study the Newton polygon jumping locus of a Mumford family in char p. Our main result says that, under a mild assumption on p, the jumping locus consists of only supersingular points and its cardinality is equal to (p^r − 1)(g − 1), where r is the degree of the defining field of the base curve of a Mumford family in char p and g is the genus of the curve. The underlying technique is the p-adic Hodge theory.

In this paper, we present a practical and provably secure two-pass authenticated key exchange protocol over ideal lattices, which is conceptually simple and has similarities to the Diffie-Hellman based protocols such as HMQV (CRYPTO 2005) and OAKE (CCS 2013). Our method does not involve other cryptographic primitives—in particular, it does not use signatures—which simplifies the protocol and enables us to base the security directly on the hardness of the ring learning with errors problem. The security is proven in the Bellare-Rogaway model with weak perfect forward secrecy in the random oracle model. We also give a one-pass variant of our two-pass protocol, which might be appealing in specific applications. Several concrete choices of parameters are provided, and a proof-of-concept implementation shows that our protocols are indeed practical.