Abstract: A methods for payment authorization (10) on mobile devices (DM) such as smartphones, tablets or any others available, which may be offline; the method for payment authorization (10) comprises the compilation of sequential steps of method (M1) of the payer (20) with method (M2) of the operational system (50) or application that constitutes a logical structure for alignment with the method (M3) of the payee (30), resulting in authenticated payment (PG) of financial transactions (TF) with assurance of “non-repudiation” through generation of a private key (51) and public key (52), as well as association of positive identification (21a) and personal identification (21b) of the payer (PG) with the mobile device (DM); said methods (M1), (M2) and (M3) are executed on mobile devices (DM) with enough processing capacity for execution of encryption algorithms and which may be used for issuing payment orders (PG), on-site or otherwise, carried out with financial resources (RF) or credit limits (LC) such as bonuses, point