Cryptography on a simplified elliptical curve

An electronic component configured to execute a cryptographic calculation and obtain a point P(X,Y) from at least one parameter t, on an elliptical curve that satisfies the equation:

• Y2=f(X) and from polynomials X1(t), X2(t), and U(t) satisfying the following equality:
• −f(X1(t))·f(X2(t))=U(t)2 in a finite field Fq, regardless of the parameter t, q satisfying the equation q=3 mod 4, said electronic component configured to:
• obtain a value of the parameter t;
• determine the point P by:
• (i) calculating X1=X1 (t), X2=X2(t) and U=U(t);
• (ii) testing whether the term f(X1) is a squared term in the finite field Fq and in this case calculating the square root of the term f(X1), the point P having X1 as abscissa and the square root of the term f(X1) as ordinate Y1;
• (iii) otherwise calculating the square root of the term f(X2), the point P having X2 as abscissa and the square root of the term f(X2) as ordinate Y2;
• and
• wherein said electronic component is further configured to use said point P in a cryptographic application selected from the group consisting of encryption or hashing or signature or authentication or identification to generate at least one of a secret key, a secret signature and an encrypted message for a user of a device, wherein the cryptographic calculation is an application of authentication or identification by a checking entity, and wherein said electronic component in obtaining the value of the parameter t is further configured to:
• /a/ generate a random value;
• /b/ obtain an encrypted value by encrypting said random value based on an encryption function using an encryption key determined from a password or identifier corresponding to the parameter;
• and
• /c/ transmit the encrypted value to the checking entity.