Efficient Big Integer Multiplication and Squaring Algorithms for Cryptographic Applications

Algorithm 7

Binary to limited Big-one converter algorithm, 𝐵𝑖𝑛2𝐿𝐵𝑂(𝐴).

Input: positive integers 𝐴=𝑎_𝑛,…,𝑎₀₂  having 𝑛+1 base 2 digits and positive integer 𝑤.

Output: 𝐶=𝑐_𝑛,…,𝑐₀₂  having 𝑛+1 digits in LCBONS and non-zero 𝑐_𝑖=𝑐_𝑖𝐿,𝑐_𝑖𝑃.

(1) Set; 𝑎₋₁←0, 𝑎_𝑛+1←0.

(2) For 𝑖 from 0 up to 𝑛+1 do the following:

  (2.1) Set 𝐿𝑒𝑛𝑔𝑡ℎ←0

  (2.2) If 𝑎_𝑖𝑎_𝑖−1=10 then do the following:

    (2.2.1) Set 𝑁𝑒𝑤𝐵𝑜←𝑇𝑟𝑢𝑒; 𝑝𝑜𝑠←𝑖.

  (2.3) While (𝑁𝑒𝑤𝐵𝑜) do the following:

    (2.3.1) Increase 𝐿𝑒𝑛𝑔𝑡ℎ and 𝑖 by 1.

    (2.3.2) If 𝑎_𝑖𝑎_𝑖−1=01 then Set 𝑁𝑒𝑤𝐵𝑜←𝐹𝑎𝑙𝑠𝑒.

    (2.3.3) If 𝐿𝑒𝑛𝑔𝑡ℎ=𝑤 then do the following:

       (2.3.3.1) Set 𝑁𝑒𝑤𝐵𝑜←𝐹𝑎𝑙𝑠𝑒 and 𝑎_𝑖−1←0.

       (2.3.3.2) Decrease 𝑖 by 1.

  (2.4) If 𝐿𝑒𝑛𝑔𝑡ℎ>0 then do the following:

    (2.4.1) Set 𝑐_𝑖𝐿←𝐿𝑒𝑛𝑔𝑡ℎ and 𝑐_𝑖𝑃←𝑝𝑜𝑠−1.

(3) Return 𝑐_𝑛,…,𝑐₀.