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ZnMoO$_4$ scintillators with a mass of $\sim$ 0.3 kg, as well asZn$^{100}$MoO$_4$ crystals enriched in the isotope $^{100}$Mo were produced forthe first time by using the low-thermal-gradient Czochralski technique. Theoptical and luminescent properties of the produced crystals were studied toestimate the progress in crystal growth quality. The low-temperature tests witha 313 g ZnMoO$_4$ and two enriched Zn$^{100}$MoO$_4$ crystals were performedaboveground in the Centre de Sciences Nucl\'eaires et de Sciences de laMati\`ere. The low background measurements with a three ZnMoO$_4$ and twoenriched detectors installed in the EDELWEISS set-up at the LaboratoireSouterrain de Modane were carried out. To optimize the light collection inZnMoO$_4$ scintillating bolometers, we have simulated the collection ofscintillation photons in a detector module for different geometries by MonteCarlo method using the GEANT4 package. Response to the 2$\nu$2$\beta$ decay of$^{100}$Mo was simulated for the enriched Zn$^{100}$MoO$_4$ detectors ofdifferent shape and mass to understand the dependence of 2$\nu$2$\beta$ decayspectra on crystal shape. We have simulated 48 Zn$^{100}$MoO$_4$ crystals witha size of {\O}60 $\times$ 40 mm installed in the EDELWEISS cryostat. Thecontribution to background from the internal radioactive contamination of thecrystals, cosmogenic activation and radioactive contamination of the set-upwere simulated. Taking into account poor time resolution of the low temperaturebolometers, we also studied contribution to background at the Q$_{2\beta}$energy of random coincidences of signals, in particular of 2$\nu$2$\beta$decay, which is one of the main sources of background in cryogenic bolometers.Methods of the randomly coinciding events rejection were developed andcompared. We have also analyzed dependence of the rejection efficiency on acryogenic detector performance.

Development of cryogenic low background detector based on enriched zinc  molybdate crystal scintillators to search for neutrinoless double beta decay  of $^{100}$Mo