Cryptography by parameterizing on elliptic curve

Electronic entity in a system for control by password comprising:

• a determination unit configured to determine on the basis of a random value r1 a point P(X,Y) on an elliptic curve in a finite field Fq, q being an integer, the elliptic curve being of equation (1):
• Ea,b(x,y):x3+ax+b=y2
• wherein first and second parameters k and k′ are obtained such that
• P=F(k,k′)
• where F is a surjective function of Fq×Fq′ in Fq,
• an encryption unit configured to encrypt the first and second parameters as a function of the password to obtain encrypted first and second parameters;
• and
• an interface unit configured to transmit said encrypted first and second parameters to the controller;
• the function F being such that, whatever z and z′ input elements of Fq, F(z,z′) is a point on the elliptic curve, and the input elements do not satisfy equation (1).

Electronic entity configured to implement a method of control which performs the following steps:

• /1/ determining on the basis of a random value r1 a point P(X,Y) on an elliptic curve, in a finite field Fq, q being an integer, of equation (1):
• Ea,b(x,y):x3+ax+b=y2
• /2/ obtaining first and second parameters k and k′, such that
• P(X,Y)=F(k,k′)
• where F is a surjective function of Fq×Fq′ in Fq,
• /3/ encrypting the first and second parameters as a function of a password to obtain encrypted first and second parameters;
• and
• /4/ transmitting said encrypted first and second parameters;
• in which the function F is such that, regardless of z and z′ input elements of Fq, F(z,z′) is a point on the elliptic curve, and the input elements do not satisfy equation (1).

A system for control by password comprising:

• a first electronic entity configured as a controller;
• and
• a second electronic entity configured as a device to be controlled by the controller;
• wherein each of the first and second electronic entities comprises:
• a determination unit configured to determine on the basis of a random value r1 a point P(X,Y) on an elliptic curve in a finite field Fq, q being an integer, the elliptic curve being of equation (1):
• Ea,b(xy):ax+b=y2
• wherein first and second parameters k and k′ are obtained such that
• P=F(k,k′)
• where F is a surjective function of Fq×Fq′ in Fq,
• an encryption unit configured to encrypt the first and second parameters as a function of the password to obtain encrypted first and second parameters;
• and
• an interface unit configured to transmit said encrypted first and second parameters to the controller;
• the function F being such that, whatever z and z′ input elements of Fq, F(z,z′) is a point on the elliptic curve, and the input elements do not satisfy equation (1).