Unified Compact ECC-AES Co-Processor with Group-Key Support for IoT Devices in Wireless Sensor Networks

. 2018 Jan 16;18(1):251. doi: 10.3390/s18010251

Algorithm 2 Montgomery ladder over projective coordinates, making explicit field operations

Require: k,

P (x, y)

Ensure:

k P

X_{1} \leftarrow x, Z_{1} \leftarrow 1, X_{2} \leftarrow x^{4} + b, Z_{2} \leftarrow x^{2} . C o m p u t e (P, 2 P)

2: for

i = m - 2 downto 0

do
3: if

k_{i} = 0

then
4:

T \leftarrow Z_{2}, Z_{2} \leftarrow {(X_{1} Z_{2} + X_{2} Z_{1})}^{2}, X_{2} \leftarrow x Z_{2} + X_{1} X_{2} Z_{1} T

T \leftarrow X_{1}, X_{1} \leftarrow X_{1}^{4} + b Z_{1}^{4}, Z_{1} \leftarrow T^{2} Z_{1}^{2}

6: else
7:

T \leftarrow Z_{1}, Z_{1} \leftarrow {(X_{1} Z_{2} + X_{2} Z_{1})}^{2}, X_{1} \leftarrow x Z_{1} + X_{1} X_{2} T Z_{2}

T \leftarrow X_{2}, X_{2} \leftarrow X_{2}^{4} + b Z_{2}^{4}, Z_{2} \leftarrow T^{2} Z_{2}^{2}

    9: end if
  10: end for
  11:

x_{3} \leftarrow X_{1} / Z_{1}

12:

y_{3} \leftarrow (x + X_{1} / Z_{1}) [(X_{1} + x Z_{1}) (X_{2} + x Z_{2}) + (x^{2} + y) (Z_{1} Z_{2})] {(x Z_{1} Z_{2})}^{- 1} + y

13: return

(x_{3}, y_{3})