Refinements of Miller's Algorithm over Weierstrass Curves Revisited

In 1986 Victor Miller described an algorithm for computing the Weil pairing in his unpublished manuscript. This algorithm has then become the core of all pairing-based cryptosystems. Many improvements of the algorithm have been presented. Most of them involve a choice of elliptic curves of a \emph{special} forms to exploit a possible twist during Tate pairing computation. Other improvements involve a reduction of the number of iterations in the Miller's algorithm. For the generic case, Blake, Murty and Xu proposed three refinements to Miller's algorithm over Weierstrass curves. Though their refinements which only reduce the total number of vertical lines in Miller's algorithm, did not give an efficient computation as other optimizations, but they can be applied for computing \emph{both} of Weil and Tate pairings on \emph{all} pairing-friendly elliptic curves. In this paper we extend the Blake-Murty-Xu's method and show how to perform an elimination of all vertical lines in Miller's algorithm during Weil/Tate pairings computation on \emph{general} elliptic curves. Experimental results show that our algorithm is faster about 25% in comparison with the original Miller's algorithm.Comment: 17 page

The Selberg zeta function for convex co-compact Schottky groups

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on ${\mathbb H}^{n+1}$: in strips parallel to the imaginary axis the zeta function is bounded by $\exp (C |s|^\delta)$ where $\delta$ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound $\exp (C |s|^{n+1})$, and it gives new bounds on the number of resonances (scattering poles) of $\Gamma \backslash {\mathbb H}^{n+1}$. The proof of this result is based on the application of holomorphic $L^2$-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider $\Gamma \backslash {\mathbb H}^{n+1}$ as the simplest model of quantum chaotic scattering. The proof of this result is based on the application of holomorphic $L^2$-techniques to the study of the determinants of the Ruelle transfer operators and on the quasi-self-similarity of limit sets

Canted antiferromagnetism in phase-pure CuMnSb

We report the low-temperature properties of phase-pure single crystals of the half-Heusler compound CuMnSb grown by means of optical float-zoning. The magnetization, specific heat, electrical resistivity, and Hall effect of our single crystals exhibit an antiferromagnetic transition at $T_{\mathrm{N}} = 55~\mathrm{K}$ and a second anomaly at a temperature $T^{*} \approx 34~\mathrm{K}$. Powder and single-crystal neutron diffraction establish an ordered magnetic moment of $(3.9\pm0.1)~\mu_{\mathrm{B}}/\mathrm{f.u.}$, consistent with the effective moment inferred from the Curie-Weiss dependence of the susceptibility. Below $T_{\mathrm{N}}$, the Mn sublattice displays commensurate type-II antiferromagnetic order with propagation vectors and magnetic moments along $\langle111\rangle$ (magnetic space group $R[I]3c$). Surprisingly, below $T^{*}$, the moments tilt away from $\langle111\rangle$ by a finite angle $\delta \approx 11^{\circ}$, forming a canted antiferromagnetic structure without uniform magnetization consistent with magnetic space group $C[B]c$. Our results establish that type-II antiferromagnetism is not the zero-temperature magnetic ground state of CuMnSb as may be expected of the face-centered cubic Mn sublattice.Comment: 14 pages, 15 figure