First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball$B$⊂$C$^{$n$}with its relative logarithmic capacity in$C$^{$n$}with respect to the same ball$B$. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of$C$^{$n$}is also proved.

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Balanced Oil and Vinegar signature schemes and the unbalanced Oil and Vinegar signature schemes are public key signature schemes based on multivariable polynomials. In this paper, we suggest a new signature scheme, which is a generalization of the Oil-Vinegar construction to improve the efficiency of the unbalanced Oil and Vinegar signature scheme. The basic idea can be described as a construction of multi-layer Oil-Vinegar construction and its generalization. We call our system a Rainbow signature scheme. We propose and implement a practical scheme, which works better than Sflash^{v_2}, in particular, in terms of signature generating time.

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