Energy/Area-Efficient Scalar Multiplication with Binary Edwards Curves for the IoT

. 2019 Feb 10;19(3):720. doi: 10.3390/s19030720

Algorithm 4 Itoh-Tsujii Inversion for Generic Binary Fields Where Additional Cycles are Required.

Input:

A (x) \in F_{2^{m}}

U = u_{0}, u_{1}, \dots u_{r - 1}

the binary representation of m,

f (x)

the irreducible polynomial of

F_{2^{m}}

Output:

C (x) = A^{- 1} (x) \mod f (x)

C (x) \leftarrow A (x)

for

i = 0

r - 1

B (x) \leftarrow C (x)

for

j = 0

2^{i}

B (x) \leftarrow B^{2} (x) \mod f (x)

end for

C (x) \leftarrow C (x) B (x) \mod f (x)

end for
for

i = r - 2

to 1 do
if

u_{i} = 1

then

B (x) \leftarrow C (x)

for

j = 0

2^{i}

B (x) \leftarrow B^{2} (x) \mod f (x)

end for

C (x) \leftarrow A (x)

for

j = 0

to i do

D (x) \leftarrow C (x)

for

k = 0

2^{j}

D (x) \leftarrow D^{2} (x) \mod f (x)

end for

C (x) \leftarrow C (x) D (x) \mod f (x)

end for

C (x) \leftarrow C (x) B (x) \mod f (x)

end if
end for

C (x) \leftarrow C^{2} (x) \mod f (x)

return

C (x)