Efficient Big Integer Multiplication and Squaring Algorithms for Cryptographic Applications

Algorithm 2

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

Input: positive integers 𝐴=𝑎_𝑛,…,𝑎_0𝑟  having 𝑛+1 base 𝑟 digits and 𝐵=𝑏_𝑚,…,𝑏_0𝑟 having 𝑚+1 base 𝑟 digits.

Output: the product 𝐴⋅𝐵=𝑐_𝑚+𝑛+1,…,𝑐_0𝑟 in base 𝑟.

    Note: (𝑢𝑣)_𝑟 are two single-precision digits in base r, indicating the result of the addition.

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

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

  (2.1) 𝑐𝑎𝑟𝑟𝑦←0.

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

    (2.2.1) Compute (𝑢𝑣)_𝑟=𝑐_𝑖+𝑗+𝑎_𝑗⋅𝑏_𝑖+𝑐𝑎𝑟𝑟𝑦, set 𝑐_𝑖+𝑗←𝑣, and 𝑐𝑎𝑟𝑟𝑦←𝑢.     // u and v are

  (2.3) 𝑐_𝑖+𝑛+1←𝑢.                                // single-precision

(3) Return 𝑐_𝑚+𝑛+1,…,𝑐₀.                             // digits in base r