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This paper discusses two encryption schemes to fix the Square scheme. Square+ uses the Plus modification of appending randomly chosen polynomials. Double-Layer Square uses a construction similar to some signature schemes, splitting the variables into two layers, one of which depends on the other.

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.

We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.