The problem of reducing an algebraic Riccati equation XCX −AX − XD + B = 0 to a unilateral quadratic matrix equation (UQME) of the kind PX + QX + R = 0 is analyzed. New reductions are introduced which enable one to prove some theoretical and computational properties. In particular we show that the structure preserving doubling algorithm of B.D.O. Anderson [Internat. J. Control, 1978] is in fact the cyclic reduction algorithm of Hockney [J. Assoc. Comput. Mach., 1965] and Buzbee, Golub, Nielson [SIAM J. Numer. Anal., 1970], applied to a suitable UQME. A new algorithm obtained by complementing our reductions with the shrink-and-shift technique of Ramaswami is presented. Finally, faster algorithms which require some non-singularity conditions, are designed. The non-singularity restriction is relaxed by introducing a suitable similarity transformation of the Hamiltonian.