#### 15.1.2 Karatsuba Multiplication

The Karatsuba multiplication algorithm is described in Knuth section 4.3.3 part A, and various other textbooks. A brief description is given here.

The inputs x and y are treated as each split into two parts of equal length (or the most significant part one limb shorter if N is odd).

     high              low
    +----------+----------+
    |    x1    |    x0    |
    +----------+----------+

    +----------+----------+
    |    y1    |    y0    |
    +----------+----------+

Let b be the power of 2 where the split occurs, i.e. if x0 is k limbs (y0 the same) then b=2^(k*mp_bits_per_limb). With that x=x1*b+x0 and y=y1*b+y0, and the following holds,

    x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0

This formula means doing only three multiplies of (N/2)x(N/2) limbs, whereas a basecase multiply of NxN limbs is equivalent to four multiplies of (N/2)x(N/2). The factors (b^2+b) etc represent the positions where the three products must be added.

     high                              low
    +--------+--------+ +--------+--------+
    |      x1*y1      | |      x0*y0      |
    +--------+--------+ +--------+--------+
              +--------+--------+
          add |      x1*y1      |
              +--------+--------+
              +--------+--------+
          add |      x0*y0      |
              +--------+--------+
              +--------+--------+
          sub | (x1-x0)*(y1-y0) |
              +--------+--------+

The term (x1-x0)*(y1-y0) is best calculated as an absolute value, and the sign used to choose to add or subtract. Notice the sum high(x0*y0)+low(x1*y1) occurs twice, so it’s possible to do 5*k limb additions, rather than 6*k, but in GMP extra function call overheads outweigh the saving.

Squaring is similar to multiplying, but with x=y the formula reduces to an equivalent with three squares,

    x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2

The final result is accumulated from those three squares the same way as for the three multiplies above. The middle term (x1-x0)^2 is now always positive.

A similar formula for both multiplying and squaring can be constructed with a middle term (x1+x0)*(y1+y0). But those sums can exceed k limbs, leading to more carry handling and additions than the form above.

Karatsuba multiplication is asymptotically an O(N^1.585) algorithm, the exponent being log(3)/log(2), representing 3 multiplies each 1/2 the size of the inputs. This is a big improvement over the basecase multiply at O(N^2) and the advantage soon overcomes the extra additions Karatsuba performs. `MUL_TOOM22_THRESHOLD` can be as little as 10 limbs. The `SQR` threshold is usually about twice the `MUL`.

The basecase algorithm will take a time of the form M(N) = a*N^2 + b*N + c and the Karatsuba algorithm K(N) = 3*M(N/2) + d*N + e, which expands to K(N) = 3/4*a*N^2 + 3/2*b*N + 3*c + d*N + e. The factor 3/4 for a means per-crossproduct speedups in the basecase code will increase the threshold since they benefit M(N) more than K(N). And conversely the 3/2 for b means linear style speedups of b will increase the threshold since they benefit K(N) more than M(N). The latter can be seen for instance when adding an optimized `mpn_sqr_diagonal` to `mpn_sqr_basecase`. Of course all speedups reduce total time, and in that sense the algorithm thresholds are merely of academic interest.