## 6 Performance Analysis and Implementation Results
### Implementation Results of Field Arithmetic  
Table 3 lists how many cycles are used for field arithmetic over \(\mathbb{F}_{p}\) and \(\mathbb{F}_{p^{2}}\) on AVR, MSP430, and ARM processors, including function call overhead. The field inversions \(\mathbb{F}_{p}\) and \(\mathbb{F}_{p^{2}}\) are the average cycles performed \(10^{4}\) times and remaining the field arithmetic is the average cycles performed \(10^{7}\) times. To evaluate the implementation of field arithmetic for curve Ted127-glv4, we compare the number of cycles for its implementation with Four\(\mathbb{Q}\), which provides the fastest implementation results to date [24]. We will now compare the number of cycles for field arithmetic on 8-bit AVR processor. The field arithmetic over \(\mathbb{F}_{p}\) for curve Ted127-glv4 on 8-bit AVR requires 198, 196, 1221, 1796, and 176,901 cycles to compute addition, subtraction, squaring, multiplication, and inversion over \(\mathbb{F}_{p}\), respectively. Similarly, the field arithmetic for Four\(\mathbb{Q}\) on AVR requires 155, 159, 1026, 1598, and 150,535 cycles to compute field addition, subtraction, squaring, multiplication, and inversion over \(\mathbb{F}_{p}\), respectively. The curve Ted127-glv4 requires 43, 37, 195, 198, and 26,366 more cycles than Four\(\mathbb{Q}\) for these operations, respectively. The field arithmetic over \(\mathbb{F}_{p^{2}}\) for curve Ted127-glv4 requires 452, 448, 4093, 6277, and 183,345 cycles to compute field addition, subtraction, squaring, multiplication, and inversion over \(\mathbb{F}_{p^{2}}\), respectively. These same operations for Four\(\mathbb{Q}\) require 384, 385, 3622, 5758, and 156,171 cycles, respectively. The curve Ted127-glv4 requires 68, 63, 471, 519, and 27,174 more cycles than Four\(\mathbb{Q}\) for these operations, respectively. In the case of the 16-bit MSP430X processor, field arithmetic over \(\mathbb{F}_{p}\) for curve Ted127-glv4 requires 120, 126, 837, 1087, and 119,629 cycles to compute field addition, subtraction, squaring, multiplication, and inversion over \(\mathbb{F}_{p}\). The same operations for Four\(\mathbb{Q}\) requires 102, 101, 927, 1027, and 131,819 cycles, respectively. The curve Ted127-glv4 requires 18, 25, and 60 more cycles than Four\(\mathbb{Q}\) to compute addition, subtraction, and multiplication, respectively. However, it saves 90 and 12,190 cycles than Four\(\mathbb{Q}\) to compute squaring and inversion over \(\mathbb{F}_{p}\), respectively. The field arithmetic over \(\mathbb{F}_{p^{2}}\) for curve Ted127-glv4 requires 266, 278, 2476, 3806, and 123,740 cycles to compute field addition, subtraction, squaring, multiplication, and inversion over \(\mathbb{F}_{p^{2}}\), respectively. These operations for Four\(\mathbb{Q}\) require 233, 231, 2391, 3624, and 135,315 cycles, respectively. The curve Ted127-glv4 requires 33, 47, 85, and 182 more cycles than Four\(\mathbb{Q}\) to compute addition, subtraction, squaring, and multiplication, respectively. It saves 11,575 cycles than Four\(\mathbb{Q}\) to compute inversion over \(\mathbb{F}_{p^{2}}\). In the 32-bit ARM Cortex-M4 processor, field arithmetic for curve Ted127-glv4 requires 55, 55, 88, 99, and 12,135 cycles to compute field addition, subtraction, squaring, multiplication, and inversion over \(\mathbb{F}_{p}\), respectively. However, Ref. [24] does not report the implementation results of field arithmetic over \(\mathbb{F}_{p}\). The field arithmetic over \(\mathbb{F}_{p^{2}}\) for curve Ted127-glv4 requires 82, 82, 196, 341, and 12,612 cycles to compute field addition, subtraction, squaring, multiplication, and inversion over \(\mathbb{F}_{p^{2}}\), respectively. These operations for Four\(\mathbb{Q}\) require 84, 86, 215, 358, and 21,056 cycles, respectively. The curve Ted127-glv4 saves 2, 4, 20, 17, and 8444 cycles than Four\(\mathbb{Q}\) to compute addition, subtraction, squaring, multiplication, and inversion over \(\mathbb{F}_{p^{2}}\), respectively.