A counterpart of the modular double for quantum superalgebra Uq(osp(1\2)) is constructed by means of supersymmetric quantum mechanics. We also construct the R-matrix operator acting in the corresponding representations, which is expressed via quantum dilogarithm.

In this paper, we first prove an explicit formula which bounds the degree of regularity of the family of HFEv (“HFE with vinegar”) and HFEv- (“HFE with vinegar and minus”) multivariate public key cryptosystems over a finite field of size q. The degree of regularity of the polynomial system derived from an HFEv- system is less than or equal to \frac{(q−1)(r+v+a−1)}{2}+2 if q is even and r+a is odd, \frac{(q−1)(r+v+a−1)}{2}+2 otherwise, where the parameters v, D, q, and a are parameters of the cryptosystem denoting respectively the number of vinegar variables, the degree of the HFE polynomial, the base field size, and the number of removed equations, and r is the “rank” paramter which in the general case is determined by D and q as r=⌊log_q (D−1)⌋+1. In particular, setting a = 0 gives us the case of HFEv where the degree of regularity is bound by \frac{(q−1)(r+v−1)}{2}+2 if q is even and r is odd, \frac{(q−1)(r+v)}{2}+2 otherwise. This formula provides the first solid theoretical estimate of the complexity of algebraic cryptanalysis of the HFEv- signature scheme, and as a corollary bounds on the complexity of a direct attack against the QUARTZ digital signature scheme. Based on some experimental evidence, we evaluate the complexity of solving QUARTZ directly using F_4/F_5 or similar Gröbner methods to be around 2^{92}.

It is well known that del Pezzo surfaces of degree 9 − n one-to-one correspond to flat En bundles over an elliptic curve. In this paper, we construct ADE-bundles over a broader class of rational surfaces that we call ADE-surfaces, and extend the above correspondence to all flat G-bundles over an elliptic curve, where G is any simply laced, simple, compact and simply connected Lie group. In what follows, we will construct G-bundles for a non-simply laced Lie group G over these rational surfaces, and extend the above correspondence to non-simply laced cases.

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A new fourth-order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced. The scheme is based on the vorticity-stream function formulation. It is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions. It is also very efficient, at every time step or Runge–Kutta stage, only two Poisson-like equations have to be solved. The Poisson-like equations are amenable to standard fast Poisson solvers usually designed for second order schemes. Detailed comparison with the second-order scheme shows the clear superiority of this new fourth-order scheme in resolving both the boundary layers and the gross features of the flow. This efficient fourth-order scheme also made it possible to compute the driven cavity flow at Reynolds number 106on a 10242grid at a reasonable cost. Fourth-order convergence is proved under mild regularity requirements. This is the first such result to our knowledge