Recent studies have shown that the entropy production rate for the master equation consists of two nonnegative terms: the adiabatic and non-adiabatic parts, where the non-adiabatic part is also known as the dissipation rate of a Boltzmann-Shannon relative entropy. In this paper, we provide some nonzero lower bounds for the relative entropy, the entropy production rate, and its adiabatic and non-adiabatic parts. These nonzero lower bounds not only reveal some novel dissipative properties for general nonequilibrium processes which are much stronger than the second law of thermodynamics, but also impose some new constraints on thermodynamic constitutive relations. Moreover, we also provide a mathematical application of these nonzero lower bounds by studying the long-time behavior of the master equation. Extensions to the Tsallis statistics are also discussed, including the nonzero lower bounds for the Tsallis-type relative entropy and its dissipation rate.

In 1989, Tsujii, Fujioka, and Hirayama proposed a family of multivariate public key cryptosystems, where the public key is given as a set of multivariate rational functions of degree 4. These cryptosystems are constructed via composition of two quadratic rational maps. In this paper, we present the cryptanalysis of this family of cryptosystems. The key point of our attack is to transform a problem of decomposition of two rational maps into a problem of decomposition of two polynomial maps. We develop a new improved 2R decomposition method and other new techniques, which allows us to find an equivalent decomposition of the rational maps to break the system completely. For the example suggested for practical applications, it is very fast to derive an equivalent private key, and it requires only a few seconds on a standard PC.

We combine the method of searching for an invariant subspace of the unbalanced Oil and Vinegar signature scheme and the Minrank method to defeat the new TTS signature scheme, which was suggested for low-cost smart card applications at CHES 2004. We show that the attack complexity is less than 2^{50}.

We show that Multivariate Public Key Cryptosystems (MPKCs) over fields of small odd prime characteristic, say 31, can be highly efficient. Indeed, at the same design security of 2^{80} under the best known attacks, odd-char MPKC is generally faster than prior MPKCs over \GF{2^k}, which are in turn faster than "traditional'' alternatives. This seemingly counter-intuitive feat is accomplished by exploiting the comparative over-abundance of small integer arithmetic resources in commodity hardware, here embodied by SSE2 or more advanced special multimedia instructions on modern x86-compatible CPUs. We explain our implementation techniques and design choices in implementing our chosen MPKC instances modulo small a odd prime. The same techniques are also applicable in modern FPGAs which often contains a large number of multipliers.

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