I really thought I had found an incredible algorithm for better multiplication times. It's not the case so I think I underevaluated the time of an instruction somewhere :D

mult.py

import random, math, time

def multiply(n, k):
	result = 0
	if k & 1 == 1:
		result = n

	for i in range(1, math.ceil(math.log(k, 2))): # log(k)
		if (k >> i) & 1 == 1: # O(1) shift
			result = result + (n << (i)) # O(log(n)) [summation] + O(1) [shift]

	return result # O(log(n) ^2) (better than O(n) ?)

def test(n, k):
	begin = time.time()
	classic = n * k
	classicTime = time.time() - begin
	begin = time.time()
	custom = multiply(n,k)
	customTime = time.time() - begin
	print(n,k,n*k, multiply(n,k), (classicTime * 1e6), (customTime * 1e6))

for i in range(1, 10):
	test(random.randrange(1, 1e100), random.randrange(1, 1e100))