Cryptography by parameterizing on elliptic curve
View Patent ↗A device is controlled by a controller on the basis of a password. A determination is made at the device or at the controller, on the basis of a random value r 1 , of a point P(X,Y) on an elliptic curve in a finite body F q , q being an integer, according to: E a,b (x, y):x 3 +ax+b=y 2 . First and second parameters k and k′ are obtained such that P(X,Y)=F(K,k′), where F is a surjective function of F q ×F q , in F q . The first and second parameters are obtained in an encrypted format by encryption in accordance with the password. The first and second encrypted parameters are then transmitted to the controller. During the control, the function F is used, such that, whatever the values of z and z′ which are input elements of F q , F(z,z′) is a point on the elliptic curve and the input elements do not satisfy the equation.
1. Method of control of a device by a controller on the basis of a password; said method comprising the following steps, at the device or at the controller:
/1/ determining on the basis of a random value r 1 a point P(X,Y) on an elliptic curve, in a finite field F q , q being an integer, of equation (1):
E a,b ( x,y ): x 3 +ax+b=y 2
/2/ obtaining first and second parameters k and k′, such that
P ( X,Y )= F ( k,k ′)
where F is a surjective function of F q ×F q′ F q ,
/3/ encrypting the first and second parameters as a function of the password to obtain encrypted first and second parameters; and
/4/ transmitting said encrypted first and second parameters to the controller;
in which the function F is such that, regardless of z and z′ input elements of F q , F(z,z′) is a point on the elliptic curve, and the input elements do not satisfy equation (1).
2. Method of control according to claim 1 , in which the function F is written:
F ( k,k ′)= f ′( k ′)+ f a,b ( k )
where f a,b is an invertible function, based on the coefficients a and b of the elliptic curve, taking an input parameter and supplying a point on the elliptic curve and
where f′ is a function generating a point on the elliptic curve as a function of a parameter; and
in which, at step /2/, the parameters k and k′ are obtained according to the following steps:
randomly generating a value of the parameter k′;
calculating a value of f′(k′);
determining a value of the parameter k according to the following equation:
k=f a,b −1 ( P ( X,Y )− f ′( k ′)).
3. Method of control according to claim 2 , in which the function f′ is written:
f ′( k ′)= k′·G
with G the generator of the set of points on the elliptic curve; and
in which a value of the parameter k is determined according to the following equation:
k=f a,b − ( P ( X,Y )− k′·G ).
4. Method of control according to claim 2 , in which the function f′ is written:
f ′( k ′)= f a,b ( k ′)
and in which a value of the parameter k is determined according to the following equation:
k=f a,b −1 ( P ( X,Y )− f a,b ( k ′)).
5. Method of control according to claim 1 , in which the function F comprises at least one invertible function f a,b , obtained by means of polynomials X 1 (k), X 2 (k), X 3 (k) and U(k) satisfying Skalba's equation:
f ( X 1 ( k ))· f ( X 2 ( k ))· f ( X 3 ( k ))= U ( k ) 2
where f is the polynomial that defines the elliptic curve E a,b .
6. Method of control according to claim 5 , in which the function f a,b makes a pair of parameters (x,y) in F q 2 correspond to a parameter u in F q , where q is equal to 2 mod 3, such that:
x
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a
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=
0.
7. Method of control according to claim 1 , further comprising the following steps:
receiving a message indicating the first and second parameters in a form encrypted with the password;
decrypting the first and second parameters and obtain parameters p and p′; and
calculating a common secret value K with the controller according to the following equation:
K=r 1 ·F ·( p,p ′)
where r 1 is the random value used at step /1/.
8. Electronic entity in a system for control by password comprising:
a determination unit configured to determine on the basis of a random value r 1 a point P(X,Y) on an elliptic curve in a finite field F q , q being an integer, the elliptic curve being of equation (1):
E a,b ( x,y ): x 3 +ax+b=y 2
wherein first and second parameters k and k′ are obtained such that
P=F ( k,k ′)
where F is a surjective function of F q ×F q′ in F q ,
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 F q , F(z,z′) is a point on the elliptic curve, and the input elements do not satisfy equation (1).
9. The electronic entity in the system of claim 8 , wherein the electronic entity comprises a device controlled by a controller on the basis of the password.
10. The electronic entity in the system of claim 8 , wherein the electronic entity comprises a controller operable to control a device on the basis of the password.
11. Electronic entity configured to implement a method of control which performs the following steps:
/1/ determining on the basis of a random value r 1 a point P(X,Y) on an elliptic curve, in a finite field F q , q being an integer, of equation (1):
E a,b ( x,y ): x 3 +ax+b=y 2
/2/ obtaining first and second parameters k and k′, such that
P ( X,Y )= F ( k,k ′)
where F is a surjective function of F q ×F q′ in F q ,
/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 F q , F(z,z′) is a point on the elliptic curve, and the input elements do not satisfy equation (1).
12. 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 r 1 a point P(X,Y) on an elliptic curve in a finite field F q , q being an integer, the elliptic curve being of equation (1):
E a,b ( xy ): ax+b=y 2
wherein first and second parameters k and k′ are obtained such that
P=F ( k,k ′)
where F is a surjective function of F q ×F q′ in F q ,
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 F q , F(z,z′) is a point on the elliptic curve, and the input elements do not satisfy equation (1).