All asymptotically fast algorithms for integer multiplication basically reduce the problem to one or several polynomial multiplications.

Polynomial multiplication can be done in subquadratic time using an evaluation-interpolation strategy: namely, to multiply two polynomials of degree less than n, we evaluate both polynomials at 2n points, compute 2n pointwise products, and recover the polynomial product by interpolation.

When one chooses roots of unity as the evaluation points, the evaluation and interpolation steps reduce to discrete Fourier transforms (forward and inverse, respectively), which can be done in O(n log n) arithmetic operations by means of the fast Fourier transform (the pointwise products require O(n) arithmetic operations).

Now, there are a couple of difficulties. How do you decompose the integers into polynomials? Is it better to use short polynomials with large coefficients, or long polynomials with small coefficients? Since the coefficients asymptotically grow with the length, you have to account for the bit complexity of the arithmetic operations in the FFTs and pointwise products (here you get recursive integer multiplications).

Which FFT algorithm should you use (there are many to choose from)?

Also, the ring of integers does not have enough roots of unity to do FFTs, so you have to lift the computation to a larger ring, such as the field of complex numbers (with numerical approximations), an appropriate finite field, or (as in the Schönhage-Strassen algorithm) a polynomial extension ring. Which choice gives the best complexity?

The improvements to integer multiplication published in the last few decades have basically been tightened analyses of such tradeoffs. I have not read this paper in detail yet, but it seems to provide a clear description in the introduction. Quote:

The idea of the new algorithm is remarkably simple. Given two n-bit integers, we split them into chunks of exponentially smaller size, say around log n bits, and thus reduce to the problem of multiplying integer polynomials of degree O(n/log n) with coeffcients of bit size O(log n). We multiply the polynomials using discrete Fourier transforms (DFTs) over C, with a working precision of O(log n) bits. To compute the DFTs, we decompose them into "short transforms" of exponentially smaller length, say length around log n, using the Cooley-Tukey method. We then use Bluestein's chirp transform to convert each short transform into a polynomial multiplication problem over C, and finally convert back to integer multiplication via Kronecker substitution. These much smaller integer multiplications are handled recursively.

When I read "integer multiplication", I think of "123 * 456". Is that what is actually being said here? If so, why would you do anything but "123 * 456" to arrive at the actual answer?

I'm sure there's a reason, but reading your explanation sounds like the solution to something much harder than just multiplying two whole numbers.

It is very easy to perform "123 * 456" but when each numbers have millions of digits it is a lot more complex. The naive algorithm require N^2 operation to multiply two numbers with N digits. This algorithm reduce this complexity.

The most simple "fast" algorithm is the karatsuba multiplication. If you want to do "12 * 34" using the classical algorithm you will compute "12 * 3" and add "12 * 4 * 10". This need 4 single digit multiplications. (the * 10 is a simple shift) Karatsuba algorithm say that you can do "13100 + (1-2)(3-4)10 + 2*4" and get the same result. This imply only 3 single digit multiplications. (again the multiplications by power of 10 are simple shifts)

Here the saving is small but if both of your numbers are N digits, you need 3 multiplication of N/2 digits that can be done recursively with the same algorithm for a final complexity around N^1.585 instead of N^2.

Instead of splitting each number in two, you can split them in more pieces and reduce further the complexity. For big but not so big numbers you use the Karatsuba algorithm or the various Toom-Cook splitting, but with very big number you switch to the Schönhage-Strassen algorithm who use the FFT. All these algorithms works by transforming the numbers in polynoms, compute their convolution and convert back to integer.

The Fürer algorithm is most efficient algorithm known for now but only theoretically, for pratical numbers we stick with Schönhage-Strassen. This paper introduce a new proof of the complexity of the Fürer algorithm as well as a few tricks to slightly improve it.

Well, sure, but the question is how you do "123 * 456". In binary you have the multiplication

   111001000
  x  1111011

and the traditional grade-school algorithm looks something like this

         111001000
        1110010000
       00000000000
      111001000000
     1110010000000
    11100100000000
   111001000000000
   ---------------
  1101101100011000
  122222221

so the answer is 1101101100011000, or 56088 in decimal. This method takes O(n^2) bitwise multiplications, where n is the number of bits in the larger of the two integers. Clearly if you're multiplying very large integers (say you're doing cryptography - the kind of computation that computers do millions of times every day) then there's clearly an interest in having faster algorithms for multiplication.

There are many ways to multiply. Some are faster than others. The typical grade school multiplication takes O(n * n) time. Other algorithms are asymptotically better than quadratic, and become more efficient when one has 1,000s of digits. Large numbers like these can occur in computer algebra problems, of which cryptography is a member.

For more details, see http://en.wikipedia.org/wiki/Multiplication_algorithm#Fast_m...

Using these types of multiplication algorithms can actually compute the result in fewer steps versus a naive algorithm when dealing with very large integers, such as integers used in cryptography.

To add, there might be applications in for example bignum or other arbitrary precision math libraries/implementations.

Makes sense. :)

>> When I read "integer multiplication", I think of "123 * 456". Is that what is actually being said here? If so, why would you do anything but "123 * 456" to arrive at the actual answer?

The paper is talking about methods to multiply integers of significant length. Think 2048bit RSA keys, or multi-million-bit numbers used in GIMPS. Of course you just use a single machine instruction for small numbers. But the hardware designers actually care about such algorithms too because it affects the number of gates and propagation delay. That said, this paper will likely only be relevant to large integer software implementations.

Some CPUs do not have a built-in mul operator, only add and sub.

As I recall that includes the Z80 instruction set. I remember this because I wanted to program my Ti-83 with assembly rather than the included basic interpreter, and was surprised to learn that the CPU in my graphing calculator could only add, subtract, and move data around.

That's a very nice explanation. Thanks!