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A recently proposed class of multivariate Public-Key Cryptosystems, the Rainbow-Like Digital Signature Schemes, in which successive sets of central variables are obtained from previous ones by solving linear equations, seem to lead to efficient schemes (TTS, TRMS, and Rainbow) that perform well on systems of low computational resources. Recently SFLASH (C^{∗−}) was broken by Dubois, Fouque, Shamir, and Stern via a differential attack. In this paper, we exhibit similar algebraic and diffential attacks, that will reduce published Rainbow-like schemes below their security levels. We will also discuss how parameters for Rainbow and TTS schemes should be chosen for practical applications.

A Tamed Transformation Method (TTM) cryptosystem was proposed by T.T.Moh in 1999. We describe how the first implementation scheme of the TTM system can be defeated. The computational complexity of our attack is 2^{33} computations on the finite field with 2^8 elements.

We analyze the torus fixed loci of Mixed Spin P fields moduli, and deduce its localization formulas with explicit factors. An algorithm toward evaluating quintic's Gromov-Witten and Fan-Jarvis-Ruan-Witten invariants is derived.

Using deformations of singular twistor spaces, a generalisation of the connected sum construction appropriate for quaternionic manifolds is introduced. This is used to construct examples of quaternionic manifolds which have no quaternionic symmetries and leads to examples of quaternionic manifolds whose twistor spaces have arbitrary algebraic dimension.