Efficient Big Integer Multiplication and Squaring Algorithms for Cryptographic Applications

Algorithm 9

Multiple-precision classical squaring, BOSQ(𝐴).

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

Output: the square 𝐴⋅𝐴=𝐴²=𝑐_2𝑛+1,…,𝑐₀₂ in base 2 representation.

(1) Compute 𝐴^′= 𝐵𝑖𝑛2𝐿𝐵𝑂-𝐿(𝐴).                     // 𝐴^′=𝑎^′_𝑛,…,𝑎^′₀

(2) For 𝑖 from 0 up to 2𝑛+1 do: 𝑐_𝑖←0.

(3) For 𝑖 from 0 up to 𝑚 do the following:

  (3.1) If 𝑎_𝑖≠0 then do the following:

    (3.1.1) Compute (𝑢𝑣)_𝑏=𝑐_2𝑖+LUT(𝑎^′_𝑖,𝑎^′_𝑖), and set 𝑐_2𝑖←𝑣, 𝑐𝑎𝑟𝑟𝑦←𝑢.

    (3.1.2) For 𝑗 from 𝑖+1 up to 𝑛 do the following:

       (3.1.2.1) Compute (𝑢𝑣)_𝑏=𝑐_𝑖+𝑗+2LUT(𝑎^′_𝑗,𝑎^′_𝑖)+𝑐𝑎𝑟𝑟𝑦.      // u is a multi-precision binary

       (3.1.2.2) Set 𝑐_𝑖+𝑗←𝑣 and 𝑐𝑎𝑟𝑟𝑦←𝑢.              // digit and v is a single-precision

                                     // binary digit

  (3.2) 𝑐_𝑖+𝑛+1←𝑢.

(4) Return 𝑐_2𝑛+1,…,𝑐₀