DETERMINISTIC MESSAGE ENCODING ON ELLIPTIC CURVES: A DUAL-CURVE APPROACH OVER BINARY AND PRIME FIELDS

This thesis aims to solve the problem of probabilistic mapping through the translation, engineering, security hardening, and empirical validation of an innovative cryptographic patent based on a dual complementary curve system defined over binary extension fields GF(2^m). Leveraging the algebraic properties of the Trace operator, the primary goal is to demonstrate how the simultaneous use of two twin curves (E0​ and E1​) allows mapping the message space with a 100% deterministic coverage, completely eliminating the need for recalculations. This thesis also demonstrate the theoretical collapse of this paradigm when applied to prime fields in odd characteristic GF(p).