Efficient Big Integer Multiplication and Squaring Algorithms for Cryptographic Applications

Algorithm 4

Recursive Karatsuba algorithm, 𝐶=KA(𝐴,𝐵).

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

Output: the product 𝐶=𝐴⋅𝐵.

(1) If 𝑛=1 then return 𝐶=𝐴×𝐵.

(2) Split 𝐴, 𝐵 into two equal parts:

    𝐴=𝐴_𝐿×𝑟^𝑛/2+𝐴_𝑅, and 𝐵=𝐵_𝐿×𝑟^𝑛/2+𝐵_𝑅.

(3) Compute the following:

    𝑑₁=KA(𝐴_𝐿,𝐵_𝐿); 𝑑₀=KA(𝐴_𝑅,𝐵_𝑅), and 𝑑_0,1=KA(𝐴_𝑅+𝐴_𝐿,𝐵_𝑅+𝐵_𝐿).

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