A paper converting or rewinding machine provides a cutting or perforation section, equipped with at least one blade arranged to work in combination with a stationary knife to provide the division of the web of paper into a plurality of sheets of measured length. The blade can be integral to a fixed support, or alternatively to a cutting roller that rotates about a longitudinal rotation axis. The blade(s) can be a perforation blade, or a cutting blade, with helical shape, and mounted to a perforation roller. The stationary knife can be mounted to a fixed roller. The blade performs a cutting, or perforation, of the web into sheets working in combination with the stationary knife. The blades and the stationary knifes are long substantially as the width of the paper. In particular, the blade has at least one weakened portion of determined size.