Elliptic Curves with the Montgomery-Form and Their Cryptographic Applications

We show that the elliptic curve cryptosystems based on the Montgomery-form E : BY 2 = X+AX+X are immune to the timingattacks by using our technique of randomized projective coordinates, while Montgomery originally introduced this type of curves for speeding up the Pollard and Elliptic Curve Methods of integer factorization [Math. Comp. Vol.48, No.177, (1987) pp.243-264]. However, it should be noted that not all the elliptic curves have the Montgomery-form, because the order of any elliptic curve with the Montgomery-form is divisible by “4”. Whereas recent ECC-standards [NIST,SEC-1] recommend that the cofactor of elliptic curve should be no greater than 4 for cryptographic applications. Therefore, we present an efficient algorithm for generating Montgomeryform elliptic curve whose cofactor is exactly “4”. Finally, we give the exact consition on the elliptic curves whether they can be represented as a Montgomery-form or not. We consider divisibility by “8” for Montgomery-form elliptic curves. We implement the proposed algorithm and give some numerical examples obtained by this.