Cryptographic method using a non-supersingular elliptic curve E in characteristic 3

摘要

A cryptographic method is provided of a type with public key over a non-supersingular elliptic curve E, determined by the simplified Weirstrass equation y²=x³+a·x²+b over a finite field GF(3ⁿ), with n being an integer greater than or equal to 1. The method includes associating an element t of said finite field with a point P′ of the elliptic field. The step of associating includes: obtaining a pre-determined quadratic non-residue η on GF(3ⁿ); obtaining a pre-determined point P=(z_P, y_P) belonging to a conic C defined by the following equation: a·η·z^2−y2+b =0; obtaining a point Q=(z_Q, y_Q), distinct from the point P belonging to the conic C and a straight line D defined by the following equation: y=t·z+y_P−t·z_P; obtaining the element ξ of GF(3ⁿ) verifying the following linear equation over GF(3): −η·ξ=(η²·z_Q)/a; and associating, with the element t of the finite field, the point P′ of the elliptic curve, for which the coordinates are defined by the pair (η·z_Q/ξ, y_Q).