  We study the elasticity of perfect 4He at zero-temperature using the
diffusion Monte Carlo method and a realistic semi-empirical pairwise potential
to describe the He-He interactions. Specifically, we calculate the value of the
elastic constants of hcp helium C_{ij} as a function of pressure up to 110 bar.
It is found that the pressure dependence of all five non-zero C_{ij} is linear
and we provide accurate parametrization of each of them. Our elastic constants
results are compared to previous variational calculations and low-temperature
measurements and in general notably good agreement is found among them.
Furthermore, we report T = 0 results for the Gruneisen parameters, sound
velocities and Debye temperature of hcp 4He. This work represents the first of
a series of computational studies aimed at thoroughly characterizing the
response of solid helium to external stress-strain.
