In continuous printing of plural sheets of recording paper, a controlling section of an image forming apparatus corrects a writing position for a first sheet of recording paper in accordance with an off-center amount detected by a carrying position sensor with respect to the first sheet of recording paper. In one embodiment, a digital signature user creates a plurality of identical \u201cpublic\u201d keys having one or more bits and a corresponding quantum mechanical one-way function. Examples further include interaction with a remote authentication service.