## 1 Introduction
Therefore, many researchers have proposed various methods to improve the efficiency of scalar multiplication. The speed-up methods for scalar multiplication can be classified into three types: methods based on speeding up the finite field exponentiation, such as comb techniques and windowing methods, scalar recoding methods, and methods that are particular to elliptic curve scalar multiplication [11]. Speed-up methods using efficiently computable endomorphisms are one type of method that are particular to elliptic curve scalar multiplication. The Gallant-Lambert-Vanstone (GLV) method proposed by [PERSON] et al. is a method for accelerating scalar multiplication by using efficiently computable endomorphisms [11]. If the cost of computing endomorphism is less than (bit-length of curve order)/3 elliptic curve point doubling (ECDBL) operations, then this method has a computational advantage. Their method reduces about half of the ECDBL operations and saves the costs of scalar multiplication by roughly 33%. Additionally, recent studies have reported that scalar multiplication methods using efficiently computable endomorphisms are significantly faster than generalized methods. The Galbraith-Lin-Scott (GLS) curves proposed by [PERSON] et al. constructed an efficiently computable endomorphism for elliptic curves defined over \(\mathbb{F}_{p^{2}}\), where \(p\) is a prime number [12]. They demonstrated that the GLV method can efficiently compute scalar multiplication on such curves. [PERSON] and [PERSON] [13] presented an efficient implementation of two-dimensional GLS curves over \(\mathbb{F}_{p^{2}}\). In 2012, [PERSON] and [PERSON] [14] proposed four-dimensional GLV-GLS curves over \(\mathbb{F}_{p^{2}}\), which generalized the GLV method and GLS curves. [PERSON] et al. [15] proposed a GLV-GLS curve over \(\mathbb{F}_{p^{2}}\), which supports the four-dimensional scalar decomposition. They reported the implementation results indicating that the four-dimensional GLV-GLS scalar multiplication reduces at most 22% of computational cost than the two-dimensional GLV method. [PERSON] et al. [16] proposed two- and four-dimensional scalar decompositions over genus 2 curves defined over \(\mathbb{F}_{p^{2}}\). [PERSON] et al. [17] introducedan eight-dimensional GLV-GLS method over genus 2 curves defined over \(\mathbb{F}_{p^{2}}\). [PERSON] et al. [18] presented the implementation results of a two-dimensional GLV method over binary GLS elliptic curves defined over \(\mathbb{F}_{2254}\). [PERSON] and [PERSON] [19] utilized the four-dimensional GLV method on genus 1 curves defined over \(\mathbb{F}_{p^{2}}\) and genus 2 curves defined over \(\mathbb{F}_{p}\). [PERSON] [20] proposed a new family of elliptic curves over \(\mathbb{F}_{p^{2}}\), called \"\(\mathbb{Q}\)-curves\". [PERSON] and [PERSON] [21] introduced a four-dimensional \(\mathbb{Q}\) curve defined over \(\mathbb{F}_{p^{2}}\), called \"Four\(\mathbb{Q}\)\". They reported the implementation results of Four\(\mathbb{Q}\) on various Intel and AMD processors. After a Four\(\mathbb{Q}\)-based approach has been proposed, many implementation results were reported considering various environments, such as AVR, MSP430, ARM, and field-programmable gate array (FPGA) devices [22; 23; 24]. An efficient Four\(\mathbb{Q}\)-based implementation on 32-bit ARM processor with the NEON single instruction multiple data (SIMD) instruction set was proposed by [PERSON] [22]. [PERSON] et al. [23] proposed a fast and compact Four\(\mathbb{Q}\)-based implementation on FPGA device. In CHES 2017, [PERSON] et al. [24] presented highly optimized implementations using curve Four\(\mathbb{Q}\) on 8-bit AVR, 16-bit MSP430, and 32-bit ARM Cortex-M4 processors, respectively. In the case of curve Ted127-glv4, [PERSON] and [PERSON] and [PERSON] et al. [14; 25] reported the implementation results on high-end processors, such as Intel Sandy Bridge, Intel Ivy Bridge, and ARM Cortex-A processors. However, efficient implementations on resource-constrained embedded devices have not been considered to date.