Efficient Big Integer Multiplication and Squaring Algorithms for Cryptographic Applications

Algorithm 5

Recursive Karatsuba squaring algorithm, 𝐶=SQKA(𝐴).

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

Output: the product 𝐶=𝐴⋅𝐴=𝐴².

(1) If 𝑛=1 then return 𝐶=𝐴×𝐴=𝐴².

(2) Split 𝐴 into two equal parts 𝐴_𝐿 and 𝐴_𝑅:

    𝐴=𝐴_𝐿×𝑟^𝑛/2+𝐴_𝑅.

(3) Compute the following:

    𝑑₁=SQKA(𝐴_𝐿); 𝑑₀=SQKA(𝐴_𝑅), and 𝑑_0,1=SQKA(𝐴_𝑅+𝐴_𝐿).

(4) Return 𝐶=𝑑₁×𝑟^𝑛+𝑑_0,1−𝑑₀−𝑑₁×𝑟^𝑛/2+𝑑₀.