The Pairing Computation on Edwards Curves

Algorithm 1

Miller's algorithm.

Input:  𝑟=∑^𝑙−1_𝑖=0𝑟_𝑖2^𝑖, where 𝑟_𝑖∈{0,1}. 𝑃∈𝐸𝔽_𝑞,𝑄∈𝐸𝔽_𝑞𝑘.

Output:  𝑓^(𝑞𝑘_𝑟^−1)/𝑟(𝑄)

(1) 𝑓←1,𝑇←𝑃

(2) for  𝑖=𝑙−2 down to 0 do do

(3)  𝑓←𝑓²⋅𝑔_𝑇,𝑇(𝑄),𝑇←2𝑇

(4)  if  𝑟_𝑖=1 then then

(5)   𝑓←𝑓⋅𝑔_𝑇,𝑃(𝑄),𝑇←𝑇+𝑃

(6)  end if

(7) end for

(8) return 𝑓^{(𝑞𝑘−1)/𝑟}