With the advent of quantum computers that showed the viability of Shor's Algorithm to factor integers, it became apparent that asymmetric cryptographic algorithms might soon become insecure. Since then, a large number of new algorithms that are conjectured to be quantum-secure have been proposed, many of which come with non-negligible trade-offs compared to current cryptosystems. Because of this, both research and standardization attempts are an ongoing effort.

In this survey, we describe one of the most promising approaches to post-quantum cryptography: cryptosystems based on supersingular isogenies. Building on top of isogenies is promising not only because they have been a well-studied topic for many decades, but also because the algorithms proposed in recent literature promise decent performance at small key sizes, especially compared to other post-quantum candidates.

After introducing the basic mathematical backgrounds required to understand the fundamental idea behind the use of supersingular isogenies as well as their relation to elliptic curves, we explain the most important protocols that have been proposed in recent years, starting with the so-called Supersingular Isogeny Diffie–Hellman. We discuss the novel approaches to well-established protocols that supersingular isogeny-based schemes introduce, analyze why it is difficult to translate certain cryptographic schemes into the supersingular isogeny case and argue that while the discussed cryptographic schemes promise to be both performant and quantum-secure, they instead introduce a trade-off in the form of increased protocol complexity.

Recurrent neural networks (RNNs) are among the most promising of the many artificial intelligence techniques now under development, showing great potential for memory, interaction, and linguistic understanding. Among the more sophisticated RNNs are long short-term memory (LSTM) and gated recurrent units (GRUs), which emulate animal brain behavior; these methods yield superior memory and learning speed because of the excellent core structure of their architectures. In this study, we attempted to make further improvements in core structure and develop a novel, compact architecture with a high learning speed. We stochastically generated 30000 RNN architectures, evaluated their performance, and selected the one most capable of memorizing long contexts with relatively few parameters. This RNN, YamRNN, had fewer parameters than LSTM and GRU by a factor of two-thirds or better and reduced the time required to achieve the same learning performance on a sequence classification task as LSTM and GRU by 80% at maximum. This novel RNN architecture is expected to be useful for addressing problems such as predictions and analyses on contextual data and also suggests that there is room for the development of better architectures.

We consider bordered complex Hadamard matrices whose core is contained in the Bose–Mesner algebra of a strongly regular graph. Examples include a complex Hadamard matrix whose core is contained in the Bose–Mesner algebra of a conference graph due to J. Wallis, F. Szöllősi, and a family of Hadamard matrices given by S. N. Singh and O. P. Dubey. In this paper, we prove that there are no other bordered complex Hadamard matrices whose core is contained in the Bose–Mesner algebra of a strongly regular graph.

One of the unsatisfactory aspects of spatial economics is the role ascribed to the agricultural sector. To study how economic activities are impacted by the falling trade costs of manufactured goods, it is convenient to assume that the agricultural sector has only one homogeneous product and that it is traded costlessly. This paper reports how these oversimplified assumptions can be improved and what new results are derived regarding the nonmanufacturing sectors. In particular, we survey how to apply this general-equilibrium approach to clarify the role of trade costs in disclosing some well-known puzzles, including the resource curse, Dutch disease, and transfer paradox.

We consider the inverse function z = g(w) of a (normalized) starlike function w = f(z) of order α on the unit disk of the complex plane with 0 < α < 1. Krzyż, Libera and Złotkiewicz obtained sharp estimates of the second and the third coefficients of g(w) in their paper (1979). Prokhorov and Szynal gave sharp estimates of the fourth coefficient of g(w) as a consequence of the solution to an extremal problem in 1981. We give a straightforward proof of the estimate of the fourth coefficient of g(w) together with explicit forms of the extremal functions.