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

1. A cryptographic method of a type with a public key over a non-supersingular elliptic curve E, determined by the simplified Weirstrass equation y 2 =x 3 +a·x 2 +b over a finite field GF(3 n ), with n being an integer greater than or equal to 1, the method comprising the following steps performed by an electronic device:

associating an element t of said finite field with a point P′ of the elliptic curve, wherein associating comprises:

obtaining a pre-determined quadratic non-residue η on GF(3 n );

obtaining a pre-determined point P=(z P , y P ) belonging to a conic C defined by the following equation: a·η·z 2 −y 2 +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(3 n ) verifying the following linear equation over GF(3): −η·ξ=(η 2 ·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 ).

using a hash function on a message m represented by a sequence of bits to produce a hashed message; and

converting the hashed message into said element t of the finite field on which the elliptic curve is defined.

2. The method according to claim 1 , wherein obtaining the element ξ of GF(3 n ) includes a computation step using the inverse of a matrix A, the elements of the matrix A being a function of a representation of the element η, and said matrix A is defined so that the following linear equation over GF(3), −η·ξ=(η 2 ·z Q )/a is equivalent to a linear equation A·X=Y, with X representing coordinates of the element ξ and Y representing coordinates of the element (η 2 ·z Q )/a.

3. A non-transitory computer-readable storage medium storing a computer program comprising a set of computer-executable instructions to implement a cryptographic method of a type with public key over a non-supersingular elliptic curve E, determined by the simplified Weirstrass equation y 2 =x 3 +a·x 2 +b over a finite field GF(3 n ), with n being an integer greater than or equal to 1, the method comprising the following steps performed by an electronic device when executing the instructions:

associating an element t of said finite field with a point P′ of the elliptic curve, wherein associating comprises:

obtaining a pre-determined quadratic non-residue η on GF(3 n );

obtaining a pre-determined point P=(z P , y P ) belonging to a conic C defined by the following equation: a·η·z 2 −y 2 +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(3 n ) verifying the following linear equation over GF(3): −η·ξ=(η 2 ·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 );

using a hash function on a message m represented by a sequence of bits to produce a hashed message; and

converting the hashed message into said element t of the finite field on which the elliptic curve is defined.

4. An electronic circuit configured to implement a cryptographic algorithm of a type with public key over a non-supersingular elliptic curve E, determined by the simplified Weirstrass equation y 2 =x 3 +a·x 2 +b over a finite field GF(3 n ), with n being an integer greater than or equal to 1, the electronic circuit comprising:

means for associating an element t of said finite field with a point P′ of the elliptic curve, wherein the means for associating comprise:

means for obtaining a pre-determined quadratic non-residue η over GF(3 n );

means for obtaining a pre-determined point P=(z P , y P ) belonging to a conic C defined by the following equation: a·η·z 2 −y 2 +b=0;

means for 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 ;

means for obtaining the element ξ of GF(3 n ) verifying the following linear equation over GF(3): −η·ξ=(η 2 ·z Q )/a; and

means for associating, with the element t of the finite field, the point P′ of the elliptic curve, the coordinates of which are defined by the pair (η·z Q /ξ, y Q );

means for using a hash function on a message m represented by a sequence of bits to produce a hashed message; and

means for converting the hashed message into said element t of the finite field on which the elliptic curve is defined.

5. A smart-card reader comprising an electronic circuit according to claim 4 .