A Novel Elliptic Curve Scalar Multiplication Algorithm against Power Analysis

Algorithm 6

New multi-scalar multiplication algorithm.

Input: 𝑃,𝑘,𝑆,𝑚

Output: 𝑄=[𝑘]𝑃+[𝑚]𝑆

  (1) Compute the NAF form of (𝑘,𝑚);

  (2) Add leading 0's in (𝑘,𝑚) so that the number of bits in

     (𝑘,𝑚) is multiple of 2, 𝑘=(𝑘_𝑙−1,𝑘_𝑙−2,…,𝑘₀)_NAF,

    𝑚=(𝑚_𝑙−1,𝑚_𝑙−2,…,𝑚₀)_NAF;

  (3) 𝑖=0;

  (4) While (𝑖<(𝑙−2)) do

    (4.1) if ((𝑘_𝑖+1,𝑘_𝑖)=(0,0) and (𝑚_𝑖+1,𝑚_𝑖) = (0, 0)) then

    (4.2) if ((𝑚_𝑖+3,𝑚_𝑖+2)=(0,0)) then

    (4.3)  set (𝑚_𝑖+3,𝑚_𝑖+2)=(0,1) and (𝑚_𝑖+1,𝑚_𝑖)

=(−2,0);

    (4.4) else if((𝑚_𝑖+3,𝑚_𝑖+2)=(0,−1)) then

    (4.5)  set (𝑚_𝑖+3,𝑚_𝑖+2)=(−1,0)and(𝑚_𝑖+1,𝑚_𝑖)

=(2,0);

    (4.6) else if ((𝑚_𝑖+3,𝑚_𝑖+2)=(0,1)) then

    (4.7)  set (𝑚_𝑖+3,𝑚_𝑖+2)=(1,0) and (𝑚_𝑖+1,𝑚_𝑖)

=(−2,0);

    (4.8) else if ((𝑚_𝑖+3,𝑚_𝑖+2)=(−1,0) then

    (4.9)  set (𝑚_𝑖+3,𝑚_𝑖+2)=(0,−1) and (𝑚_𝑖+1,𝑚_𝑖)

=(−2,0);

    (4.10) else

    (4.11)  set (𝑚_𝑖+3,𝑚_𝑖+2)=(0,1) and (𝑚_𝑖+1,𝑚_𝑖)

=(2,0);

    (4.12) else

    (4.13) 𝑖=𝑖+2;

  (5) Pre-computation

 𝑇[0,1]=𝑆;𝑇[0,2]=2𝑆;𝑇[0,4]=4𝑆;𝑇[1,0]=𝑃

 ;𝑇[2,0]=2𝑃;𝑇[1,1]=𝑃+𝑆;𝑇[1,2]=𝑃+2𝑆;

 𝑇[1,−1]=𝑃−𝑆;𝑇[1,−2]=𝑃−2𝑆;𝑇[2,1]=2𝑃+𝑆;

𝑇[2,2]=2𝑃+2𝑆;𝑇[2,−1]=2𝑃−𝑆;

  𝑇[2,−2]=2𝑃−2𝑆;

  (6) 𝑅=𝑇[(𝑘_𝑙−1,𝑘_𝑙−2),(𝑚_𝑙−1,𝑚_𝑙−2)];

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

   (7.1) 𝑅=2𝑅;

    (7.2) 𝑅=2𝑅;

    (7.3) if ((𝑘_𝑖−1,𝑘_𝑖−2)<0) then

    (7.4) 𝑅=𝑅−𝑇[−(𝑘_𝑖−1,𝑘_𝑖−2),−(𝑚_𝑖−1,𝑚_𝑖−2)];

    (7.5) else if ((𝑘_𝑖−1,𝑘_𝑖−2)==0) then

    (7.6)  if ((𝑚_𝑖−1,𝑚_𝑖−2)<0) then

    (7.7 )  𝑅=𝑅−𝑇[(𝑘_𝑖−1,𝑘_𝑖−2),−(𝑚_𝑖−1,𝑚_𝑖−2)];

    (7.8) else

    (7.9)  𝑅=𝑅+𝑇[(𝑘_𝑖−1,𝑘_𝑖−2),(𝑚_𝑖−1,𝑚_𝑖−2)];

    (7.10) else

    (7.11) 𝑅=𝑅+𝑇[(𝑘_𝑖−1,𝑘_𝑖−2),(𝑚_𝑖−1,𝑚_𝑖−2)];

  (8) Return (𝑅).