A Novel Elliptic Curve Scalar Multiplication Algorithm against Power Analysis

Algorithm 7

DPA-resistant scalar multiplication algorithm.

Input: P, k

Output: Q = [k]P

 (1) Select a random r, 𝑘1=𝑘−𝑟,𝑘2=𝑟;

 (2) Compute the NAF form of (𝑘1,𝑘2);

 (3) Add leading 0's in (𝑘1,𝑘2) so that the number of bits in

     (𝑘1,𝑘2) is multiple of 2, 𝑘1=(𝑘1_𝑙−1,𝑘1_𝑙−2,…,𝑘1₀)_NAF,

     𝑘2=(𝑘2_𝑙−1,𝑘2_𝑙−2,…,𝑘2₀)_NAF;

 (4) 𝑖=0;

 (5) While (𝑖<(𝑙−2)) do

   (5.1) set 𝑐=(𝑘1_𝑖+1,𝑘1_𝑖)+(𝑘2_𝑖+1,𝑘2_𝑖);

    (5.2) if (𝑐=0) then

    (5.3) if ((𝑘2_𝑖+3,𝑘2_𝑖+2)>0) then

    (5.4)  set 𝑘2_𝑖+3,𝑘2_𝑖+2−=(0,1) and (𝑘2_𝑖+1,𝑘2_𝑖)

=(−2,0);

    (5.5)  else

    (5.6)  set 𝑘2_𝑖+3,𝑘2_𝑖+2+=(0,1) and (𝑘2_𝑖+1,𝑘2_𝑖)

=(2,0);

    (5.7) else

    (5.8) 𝑖=𝑖+2;

  (6) Pre-computation

     𝑇[1]=𝑃;𝑇[2]=2𝑃;𝑇[3]=3𝑃;𝑇[4]=4𝑃;

  (7) if ((𝑘1_𝑙−1,𝑘1_𝑙−2)+(𝑘2_𝑙−1,𝑘2_𝑙−2)=0) then

     𝑅=𝑂;

    else

𝑅=𝑇[(𝑘1_𝑙−1,𝑘1_𝑙−2)+(𝑘2_𝑙−1,𝑘2_𝑙−2)];

  (8) For 𝑖 from 𝑙−2 downto 0 𝑖=𝑖−2 do

    (8.1) 𝑐=(𝑘1_𝑖−1,𝑘1_𝑖−2)+(𝑘2_𝑖−1,𝑘2_𝑖−2);

    (8.2) 𝑅=2𝑅;

    (8.3) 𝑅=2𝑅;

    (8.4) if (𝑐<0) then

    (8.5) 𝑅=𝑅−𝑇[−𝑐];

    (8.6) else

    (8.7) 𝑅=𝑅+𝑇[𝑐];

  (9) Return (𝑅).