Dynamics of low-temperature water in nanostructured materials is important to a variety of phenomena, ranging from transport in cement and asphaltene, to conformational dynamics of proteins in "crowded" cellular environments, survival of microorganisms at very low temperatures, and diffusion in nanogeoscience. Using silicon-carbide nanotubes as a prototype of nanostructured materials, extensive molecular dynamics simulations were carried out to study the cage correlation function C(t) and self-diffusivity D of supercooled water in the nanotubes. C(t), which measures changes in the atomic surroundings inside the nanotube, follows the Kohlrausch-Williams-Watts law, C(t)∼exp[-(t/τ)^{β}], where τ is a relaxation time and β is a topological exponent. For the temperature range 220K<T≤273 K, we find β≃0.438, in excellent agreement with and confirming the prediction by Phillips [Rep. Prog. Phys. 59, 1133 (1996)]RPPHAG0034-488510.1088/0034-4885/59/9/003, β=3/7. The self-diffusivity manifests a transition around 230 K, very close to 228 K, the temperature at which a fragile-to-strong dynamic crossover is supposed to happen. Thus the results indicate that water does not freeze in the nanotube over the studied temperature range, and that the Stokes-Einstein relation breaks down.