Reconfigurable Architecture for Elliptic Curve Cryptography Using FPGA

Algorithm 3

Montgomery scalar multiplication algorithm.

  Input: 𝑘=(𝑘_𝑛−1,𝑘_𝑛−2,…,𝑘₁,𝑘₀)2 with 𝑘_𝑛−1=1,

     𝑃𝑥,𝑦𝜀𝐸 (GF(2^𝑚)).

  Output: 𝑄=𝐾𝑃.

(1) Set 𝑥₁=𝑥, 𝑧₁=1, 𝑥₂=𝑥⁴+𝑏, 𝑧₂=𝑥²;

(2) for 𝑖=𝑛−2 to 0 do

(3) if 𝑘_𝑖=1 then

(4)  𝑥₁,𝑧₁←𝑀𝑎𝑑𝑑 (𝑥₁,𝑧₁,𝑥₂,𝑧₂,𝑥);

    (𝑥₂,𝑦₂)←𝑀𝑑𝑜𝑢𝑏𝑙𝑒 (𝑥₂,𝑧₂,𝑏);

(5) else

(6)  (𝑥₂,𝑦₂)←𝑀𝑎𝑑𝑑 (𝑥₂,𝑧₂,𝑥₁,𝑧₁,𝑥);

    (𝑥₁,𝑦₁)←𝑀𝑑𝑜𝑢𝑏𝑙𝑒 (𝑥₁,𝑧₁,𝑏);

(7) end if

(8) end for

(9) 𝑄←𝑀𝑥𝑦(𝑥₁,𝑧₁,𝑥₂,𝑧₂,𝑥,𝑦);

(10) return 𝑄;