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2. Trachtenberg system - Wikipedia

   en.wikipedia.org/wiki/Trachtenberg_system

   Trachtenberg defined this algorithm with a kind of pairwise multiplication where two digits are multiplied by one digit, essentially only keeping the middle digit of the result. By performing the above algorithm with this pairwise multiplication, even fewer temporary results need to be held. Example:

3. Multiplication algorithm - Wikipedia

   en.wikipedia.org/wiki/Multiplication_algorithm

   To multiply two numbers with n digits using this method, one needs about n 2 operations. More formally, multiplying two n-digit numbers using long multiplication requires Θ(n 2) single-digit operations (additions and multiplications).

4. Karatsuba algorithm - Wikipedia

   en.wikipedia.org/wiki/Karatsuba_algorithm

   Since the additions, subtractions, and digit shifts (multiplications by powers of B) in Karatsuba's basic step take time proportional to n, their cost becomes negligible as n increases. More precisely, if T(n) denotes the total number of elementary operations that the algorithm performs when multiplying two n-digit numbers, then

5. Lattice multiplication - Wikipedia

   en.wikipedia.org/wiki/Lattice_multiplication

   [2] Step 1. After all the cells are filled in this manner, the digits in each diagonal are summed, working from the bottom right diagonal to the top left. Each diagonal sum is written where the diagonal ends. If the sum contains more than one digit, the value of the tens place is carried into the next diagonal (see Step 2). Step 2. Numbers are ...

6. Napier's bones - Wikipedia

   en.wikipedia.org/wiki/Napier's_bones

   When multiplying a multi-digit number by another multi-digit number, the larger number is set up on the rods in the frame. An intermediate result is produced by the device for multiplication by each of the digits of the smaller number. These are written down and the final result is calculated by pen and paper.

7. Schönhage–Strassen algorithm - Wikipedia

   en.wikipedia.org/wiki/Schönhage–Strassen...

   The run-time bit complexity to multiply two n-digit numbers using the algorithm is (⁡ ⁡ ⁡) in big O notation. The Schönhage–Strassen algorithm was the asymptotically fastest multiplication method known from 1971 until 2007.