Efficient Big Integer Multiplication and Squaring Algorithms for Cryptographic Applications

Algorithm 1

Right-to-left binary exponentiation, 𝐛=𝐚^𝐱.

Input: positive integers 𝑎 and 𝑥>1.

Output: 𝑏=𝑎^𝑥

(1) Set 𝑏←1 and 𝑠←𝑎.

(2) While 𝑥≠0 do the following:

  (2.1) If 𝑥 is odd then 𝑏←𝑏×𝑠.       // (𝑘/2) multiplication if  𝑘=log₂𝑏.

  (2.2) Set 𝑥←⌊𝑥/2⌋.

  (2.3) If 𝑥≠0   then 𝑠←𝑠×𝑠.       // (𝑘−1)  multiplication.

(3) Return 𝑏.