Malaysian Journal of Mathematical Sciences, September 2024, Vol. 18, No. 3
Fast Endomorphisms in Integer Sub-Decomposition Method on Secp192k1
Antony, S. N. F. M. A., Yion, C. H. K., Kamarulhaili, H., Ariffin, M. R. K., and Yunos, F.
Corresponding Email: [email protected]
Received date: 17 November 2023
Accepted date: 25 March 2024
Abstract:
Elliptic curve cryptography involves numerous scalar multiplications, incurring high operational costs. In view of this, fast endomorphism is used to represent scalar multiplications, $kP$ on elliptic curves. In the past, techniques such as Gallant-Lambert-Vanstone (GLV) method and Integer Sub-Decomposition (ISD) method have been proposed to reduce the cost of scalar multiplication on elliptic curves by using fast endomorphism. The GLV method employs a single-layer decomposition, breaking $k$ into $k_{1}$ and $k_{2}$, while the ISD method uses a bilayer decomposition. The existence of fast endomorphisms which are constructed based on the concept of isogeny increase the computational efficiency of the GLV approach and reduce the operation count on the ISD method. This paper embeds the fast endomorphisms in the scalar multiplications on one of the family of elliptic curves with j-invariant 0, $E_{0}$, which is the 192-bit Koblitz curve (Secp192k1). The performance of the ISD method in computing certain scalar multiplications on Secp192k1 in conjunction with fast endomorphisms and other various techniques such as binary representation, NAF representation, w-NAF and sliding windows are computed. The results demonstrated that the ISD method together with fast endomorphism, yields the most promising outcomes. This underscores the advantages of using fast endomorphisms in the ISD method on $E_{0}$.
Keywords: elliptic curve scalar multiplication; fast endomorphism; GLV method; ISD method; Secp192k1
