We discuss approximations of the vertex coupling on a star-shaped quantum graph of n edges in the singular case when the wave functions are not continuous at the vertex and no edge-permutation symmetry is present. It is shown that the CheonShigehara technique using interactions with nonlinearly scaled couplings yields a 2n-parameter family of boundary conditions in the sense of norm resolvent topology. Moreover, using graphs with additional edges, one can approximate the {n+1\choose 2}-parameter family of all time-reversal invariant couplings.

In 2006, the inventors of TRMC public key cryptosystem proposed a new variant of TRMC, TRMC-4, which can resist the existing attack, in particular, the Joux et al attack. In this paper, we show that the new version is vulnerable to attack via the linearization equations (LE) method. For any given valid ciphertext and its corresponding TRMC-4 public key, we can derive the corresponding plaintext within 2^{24} F_{2^8}-operations, after performing once for the public key a computation of complexity less than 2^{34}. Our results are confirmed by computer experiments.

The SMS4 block cipher is part of the Chinese WAPI wireless standard. This paper describes the specification and offers a specification for a toy version called simplified SMS4 (S-SMS4). We explore algebraic attacks on SMS4 and S-SMS4 using Gröbner basis attacks on equation systems over GF(2) and GF(2^8), as well as attacks using a SAT solver derived from the GF(2) model. A comparison of SAT and Gröbner basis attacks is provided.

In this paper, we propose an algebraic broadcast attack against NTRU, which recovers a single message encrypted multiple times using different NTRU public keys. Namely, when a message is broadcasted, under some reasonable assumptions, our attack can be completed in polynomial time and space. To the best of our knowledge, this is the first successful broadcast attack against NTRU.

The SimpleMatrix encryption scheme as proposed by Tao et al. \cite{TD13} is one of the very few existing approaches to create a secure and efficient encryption scheme on the basis of multivariate polynomials. However, in its basic version, decryption failures occur with non-negligible probability. Although this problem has been addressed in several papers \cite{DP14,TX15}, a general solution to it is still missing. In this paper we propose an improved version of the SimpleMatrix scheme, which eliminates decryption failures completely and therefore solves the biggest problem of the SimpleMatrix encryption scheme. Additionally, we propose a second version of the scheme, which reduces the blow-up factor between plain and ciphertext size to a value arbitrary close to 1.