We investigate the gauge boson propagator in three dimensional compact Abelian gauge model in the Landau gauge at finite temperature. The presence of the monopole plasma in the confinement phase leads to appearance of an anomalous dimension in the momentum dependence of the propagator. The anomalous dimension as well as an appropriate ratio of photon wave function renormalization constants with and without monopoles are observed to be order parameters for the deconfinement phase transition. We discuss the relation between our results and the confining properties of the gluon propagator in non-Abelian gauge theories. Here, we want to elaborate on cQED 3 as a toy model of confinement. Indeed, this has been the first non-trivial case in which confinement of electrically charged particles was understood analytically [1]. Confinement is caused here by a plasma of monopoles which emerge due to the compactness of the gauge field. Other common features of the two theories are the existence of a mass gap and of a confinement-deconfinement phase transition at some non-zero temperature. According to universality arguments [5] the phase transition of cQED 3 is expected to be of Kosterlitz-Thouless type [6].In QCD 4 , the deconfinement phase transition is widely believed to be caused by loss of monopole condensation (for a review see Ref. [7]) within the effective dual superconductor approach [8]. Studying the dynamics of the monopole current inside gluodynamics, monopole decondensation at the critical temperature is appearing as de-percolation, i.e. the decay of the infrared, percolating monopole cluster into short monopole loops [9]. This change of vacuum structure has a dimensionally reduced analog in the 3D monopole-antimonopole pair binding which has been observed in cQED 3 [10,11].At present, the gluon propagator in QCD 4 is under intensive study. The analogies mentioned before encouraged us to study the similarities between the gauge boson propagators in both theories. In order to fix the role of the monopole plasma in cQED 3 , not just for confinement of external charges but also for the non-perturbative modification of the gauge boson propagator, we consider it in the confinement and the deconfined phases. On the other hand, on the lattice at any temperature we are able to separate the monopole contribution to the propagator by means of eq. (2) below.We have chosen the Landau gauge since it has been adopted in most of the investigations of the gauge boson propagators in QCD [12,13] and QED [14,15]. In order to avoid the problem of Gribov copies [16], the alternative Laplacian gauge has been used recently [17]. The Coulomb gauge, augmented by a suitable global gauge in each time slice (minimal Coulomb gauge) has been advocated both analytically [18] and numerically [19].The numerical lattice results for gluodynamics show that the propagator for all these gauges in momentum space is less singular than p −2 in the immediate vicinity of p 2 = 0. Moreover, the results for the propagator at zero momentum are ranging from ...