Efficient Big Integer Multiplication and Squaring Algorithms for Cryptographic Applications

Algorithm 8

Multiple-precision classical multiplication, BOCM(𝐴,𝐵).

Input: positive integers 𝐴=𝑎_𝑛,…,𝑎₀₂ and 𝐵=𝑏_𝑚,…,𝑏₀₂  having  𝑛+1 base 2 digits.

Output: the product 𝐴⋅𝐵=𝑐_𝑚+𝑛+1,…,𝑐₀₂ in base 2 representation.

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

(2) For 𝑖 from 0 up to 𝑚+𝑛+1 do: 𝑐_𝑖←0.                   // 𝐵^′=𝑏^′_𝑛,…,𝑏^′₀

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

  (3.1) Set 𝑐𝑎𝑟𝑟𝑦←0

  (3.2) If 𝑏_𝑖≠0 then do the following:

    (3.2.1) For 𝑗 from 0 up to 𝑛 do the following:

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

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

  (3.3) 𝑐_𝑖+𝑛+1←𝑢.

(4) Return 𝑐_2𝑛+1,…𝑐₀.