Patent ID: 6876745

Claim:
A method of implementing an elliptic curve cryptographic operation in a cryptographic apparatus implementing an elliptic curve cryptography in a finite field of characteristic 2 (or an extension field of “2”), in which said elliptic curve is given by y 2 +xy=ax 2 +b and in which x and y are variables in an x-y coordinate system, a and b are parameters, addition of points P 1 (x 1 , y 1 ) and P 2 (x 2 , y 2 ) on said elliptic curve composed of points defined by individual coordinate components is presumed to be represented by P 3 (x 3 , y 3 ) with subtraction of said points P 1 (x 1 , y 1 ) and P 2 (x 2 , y 2 ) being presumed to be represented by P 4 (x 4 , y 4 ), said method comprising the steps performed by said cryptographic apparatus, of: inputting the coordinate component x 1 ; transforming the inputted coordinate component x 1 into x-coordinates and z-coordinates [X 1 , Z 1 ] of a projective space where z is a variable of a projective space where z is a variable in the z-coordinate; storing said coordinates [X 1 , Z 1 ] of said projective space; transforming the coordinate component x 2 into coordinates [X 2 , Z 2 ] of said projective space; storing the projective coordinates [X 2 , Z 2 ]; transforming the coordinate component x 4 into coordinates [X 4 , Z 4 ] of said projective space; storing the coordinates [X 4 , Z 4 ]; determining projective coordinates [X 3 , Z 3 ] from the stored projective coordinates [X 1 , Z 1 ], [X 2 , Z 2 ] and [X 4 , Z 4 ]; transforming said projective coordinates [X 3 , Z 3 ] into the coordinate component x 3 ; and outputting said coordinate component x 3 , whereby scalar multiplication of said point P 1 (x 1 , y 1 ) is determined; generating a random number k; storing said generated random number k; transforming the x-coordinates into projective coordinates to thereby derive projective coordinates [k 2 x, k] through arithmetic operation of individual coordinate components of said projective space and said stored random number k.