The present disclosure relates to a composition for permanently reshaping keratin fibers, such as the hair, comprising ammonium bicarbonate and at least one other alkaline agent chosen from aqueous ammonia and ammonium salts other than ammonium bicarbonate, with an ammonium molar concentration ratio of greater than 2.1, at least one non-cyclic cationic polymer, and at least one reducing agent, wherein, when aqueous ammonia is present, it is present in an amount of less than 3% by weight. Also disclosed herein is permanent-reshaping process comprising the application of this reducing composition in a first stage, followed, after a leave-in time, by the application of a fixing composition and rinsing of the keratin fibers thus treated.The present disclosure further relates to a multi-compartment device or kit comprising this reducing composition.