Efficient Big Integer Multiplication and Squaring Algorithms for Cryptographic Applications

Algorithm 3

Multiple-precision classical squaring, SQ(𝐴).

Input: positive integer 𝐴=𝑎_𝑛,…,𝑎_0𝑟  having 𝑛+1 base 𝑟 digits.

Output: the square 𝐴⋅𝐴=𝐴²=𝑐_2𝑛+1,…,𝑐_0𝑟 in base 𝑟 representation.

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

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

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

  (2.1) Compute (𝑢𝑣)_𝑟=𝑐_2𝑖+𝑎_𝑖⋅𝑎_𝑖, set 𝑐_2𝑖←𝑣, and 𝑐𝑎𝑟𝑟𝑦←𝑢.

  (2.2) For 𝑗 from 𝑖+1 up to 𝑛 do the following:                    // v  is a single-

    (2.2.1) Compute (𝑢𝑣)_𝑟=𝑐_𝑖+𝑗+2𝑎_𝑗⋅𝑎_𝑖+𝑐𝑎𝑟𝑟𝑦, set 𝑐_𝑖+𝑗←𝑣, and 𝑐𝑎𝑟𝑟𝑦←𝑢.    // precision digit and u

  (2.3) 𝑐_𝑖+𝑛+1←𝑢.                                 // is a multi-precision

(3) Return 𝑐_2𝑛+1,…,𝑐₀.                             // digit in base r