Apparatus and a method for calculating a multiple of a point an elliptic curve

1. A cryptographic method for calculating a multiple of a point on an elliptic curve from the right to the left by repeated point addition and point doubling in an elliptic curve cryptosystem, the method comprising:

evaluating each point doubling by a micro-processor by taking as input a set of coordinates comprising projective coordinates T 1 , T 2 , T 3 and at least one further coordinate T 4 ;

evaluating each point addition by the micro-processor by taking as input the projective coordinates T 1 , T 2 , T 3 ; and

storing by the micro-processor the at least one further coordinate T 4 in a memory device for use in the next point doubling,

wherein output coordinates of a point doubling is used as input coordinates for the next point doubling,

wherein the point doubling is calculated by:

obtaining the values of the projective coordinates T 1 , T 2 , T 3 and the value of the further coordinate, T 4 , initialized to T 4 =aT 3 4 , where a is a first parameter of the elliptic curve;

calculating intermediate variables:

U=T 1 2 ;

V=T 2 2 ;

M= 3 U+T 4 ;

W=V 2 ; and

S= 2(( T 1 +V ) 2 −U−W );

T 1 =M 2 −2 S;

calculating new value of T 2 ;

T 2 =M ( S−T 1 )−8 W ; and

outputting at least the values of coordinates T 1 , T 2 , T 3 .

2. The method of claim 1 , wherein the elliptic curve is given by a Weierstrass equation E IK : Y 2 =X 3 +aXZ 4 +bZ 6 , with a characteristic different from 2 and 3, wherein b is a second parameter of the elliptic curve.

3. The method of claim 1 , wherein the scalar multiplication is performed using a Non-Adjacent Form of the scalar.

4. The method of claim 1 , wherein the scalar multiplication is performed using a binary representation of the scalar.

5. The method of claim 1 , wherein the point doubling is performed using modified Jacobian coordinates and the point addition is performed using Jacobian coordinates.

6. A device for calculating a multiple of a point on an elliptic curve from the right to the left in an elliptic curve cryptosystem, the device comprising memory and a micro-processor configured to:

perform repeated point doubling and point addition;

evaluate each point doubling by taking as input a set of coordinates comprising projective coordinates T 1 , T 2 , T 3 and at least one further coordinate T 4 ;

evaluate each point addition by taking as input the set of projective coordinates T 1 , T 2 , T 3 ; and

store the at least one further coordinate T 4 in the memory for use in the next point doubling,

wherein output coordinates of a point doubling is used as input coordinates for the next point doubling,

wherein the point doubling is calculated by:

obtaining the values of the projective coordinates T 1 , T 2 , T 3 and the value of the further coordinate, T 4 , initialized to T 4 =aT 3 4 , where a is a first parameter of the elliptic curve;

calculating intermediate variables:

U=T 1 2 ;

V=T 2 2 ;

M= 3 U+T 4 ;

W=V 2 ; and

S= 2(( T 1 +V ) 2 −U−W );

calculating new values of T 3 and T 4 :

T 3 =2 T 2 T 3 ;

T 4 =16 WT 4 ,

calculating new value of T 1 :

T 1 =M 2 −2 S;

calculating new value of T 2 :

T 2 =M ( S−T 1 )−8 W ; and

outputting at least the values of coordinates T 1 , T 2 , T 3 .

7. A non-transitory medium storing a computer program that, when executed on a micro-processor, performs a cryptographic method for calculating a multiple of a point on an elliptic curve from the right to the left by repeated point doubling and point addition in an elliptic curve cryptosystem, the method comprising:

evaluating each point doubling by the micro-processor by taking input a set of coordinates comprising projective coordinates T 1 , T 2 , T 3 and at least one further coordinate T 4 ;

evaluating each point addition by the micro-processor by taking as input the projective coordinates T 1 , T 2 , T 3 ; and

storing by the micro-processor the at least one further coordinate T 4 in a memory device for use in the next point doubling,

wherein output coordinates of a point doubling is used as input coordinates for the next point doubling,

wherein the point doubling is calculated by:

obtaining the values of the projective coordinates T 1 , T 2 , T 3 and the value of the further coordinate, T 4 , initialized to T 4 =aT 3 4 , where a is a first parameter of the elliptic curve;

calculating intermediate variables:

U=T 1 2 ;

V=T 2 2 ;

M= 3 U+T 4 ;

W=V 2 ; and

S= 2(( T 1 +V ) 2 −U−W );

calculating new values of T 3 and T 4 :

T 3 =2 T 2 T 3 ;

T 4 =16 WT 4 ,

calculating new value of T 1 :

T 1 =M 2 −2 S;

calculating new value of T 2 :

T 2 =M ( S−T 1 )−8 W ; and

outputting at least the values of coordinates T 1 , T 2 , T 3 .