Malaysian Journal of Mathematical Sciences, August 2019, Vol. 13(S)
Special Issue: The 6th International Cryptology and Information Security Conference (CRYPTOLOGY2018)
Construction of Endomorphisms for the ISD Method on Elliptic Curves with j-invariant 1728
Antony, S. N. F. M. A. and Kamarulhaili, H.
Corresponding Email: [email protected]
Received date: -
Accepted date: -
Abstract:
In this study, we construct the efficiently computable endomorphisms on elliptic curves with j-invariant 1728, to accelerate the computation of the ISD method. The ISD method computed scalar multiplication on elliptic curves where it requires three endomorphisms to accomplish. However, the original ISD method only able to solve integer multiplications since their endomorphisms are defined over \(\mathbb{Z}\). Besides, the endomorphisms defined in the original ISD method are not efficiently computable. We extend the study by defining the endomorphisms in the ISD method over the \(\mathbb{Q}(\sqrt{-d})\) so that it can solve complex multiplications. Elliptic curves with j-invariant 1728 are defined over \(\mathbb{Q}(i)\), where its discriminant is given as \(D=-4\), with a unique maximal order. The maximal order satisfies a polynomial of degree two, which represents the minimal polynomial for the first efficiently computable endomorphism. Meanwhile, we choose the other two endomorphisms to belong to \(\mathbb{Q}(i)\) as well.
Keywords: Efficient endomorphism, elliptic curve scalar multiplication, Integer Sub-Decomposition method, j-invariant 1728, quadratic field
