We present a fast algorithm for generating full sky, high resolution ($sim 5$) simulations of the CMB anisotropy pattern. We also discuss the inverse problem, that of evaluating from such a map the full set of $a_{ell m}$s and the spectral coefficients $C_ell$. We show that using an Equidistant Cylindrical Projection of the sky substantially speeds up the calculations. Thus, generating and/or inverting a full sky, high resolution map can be easily achieved with present day computer technology.
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