This survey article gives an account of quasiconformal extensions of univalent functions with its motivational background from Teichmüller theory and classical and modern approaches based on Loewner theory.

We study conformal embeddings of a noncompact Riemann surface of finite genus into compact Riemann surfaces of the same genus and show some of the close relationships between the classical theory of univalent functions and our results. Some new problems are also discussed. This article partially intends to introduce our results and to invite the function-theorists on plane domains to the topics on Riemann surfaces.

In this paper we investigate normalized conformal mappings of the exterior of the reciprocal of the Multibrot set and analyze the growth of the denominator of the coefficients. Our inequality improves Ewing and Schober's result which was presented in [6]. We use the coefficient formula of [13]. The straightforward adaptation of the proof in this paper slightly improves the main theorem of [12].

For a real constant b, we give sharp estimates of log |f(z)/z|+b arg[f(z)/z] for subclasses of normalized univalent functions f on the unit disk.

Let Λ be any integral lattice in Euclidean space. It has been shown that for every integer n>0, there is a hypersphere that passes through exactly n points of Λ. Using this result, we introduce new lattice invariants and give some computational results related to two-dimensional Euclidean lattices of class number one.

Lossy identification schemes derive tightly secure signature schemes via the Fiat–Shamir transformation. There exist several instantiations of lossy identification schemes by using several cryptographic assumptions. In this paper, we propose a new construction of the lossy identification scheme from the decisional RSA assumption which are introduced by Groth. Our lossy identification scheme has an efficient response algorithm because it requires no modular exponentiation.

Inspired by the work of Ghadafi and Groth (ASIACRYPT 2017) on a certain type of computational hardness assumptions in cyclic groups (which they call ``target assumptions''), we initiate an analogous work on another type of hardness assumptions, namely the ``knowledge-of-exponent'' assumptions (KEAs). Originally introduced by Damgard to construct practical encryption schemes secure against chosen ciphertext attacks, KEAs have subsequently been used primarily to construct succinct non-interactive arguments of knowledge (SNARKs), and proved to be inherent to such constructions. Since SNARKs (and their zero-knowledge variant, zk-SNARKs) are already used in practice in such systems as the Zcash digital currency, it can be expected that the use of KEAs will increase in the future, which makes it important to have a good understanding of those assumptions. Using a proof technique first introduced by Bellare and Palacio (but acknowledged by them as being due to Halevi), we first investigate the internal structure of the q-power knowledge-of-exponent (q-PKE) family of assumptions introduced by Groth, which is thus far the most general variant of KEAs. We then introduce a generalisation of the q-PKE family, and show that it can be simplified.