Patent ID: 8582758

Claim:
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 .