Take a look at the IV characteristic graph of the diode. In the reverse bias region the current has almost a constant value with a slight slope until the breakdown voltage. In the reverse bias region it is the minority carriers which contribute to the reverse saturation current. This flow is diffusive and depends on the temperature and not on the applied voltage. The minority carriers are thermally generated and hence the reverse saturation current remains constant until the breakdown voltage. Meanwhile the majority flow is prohibited due to the widened depletion region. Hence the behaviour.
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