Squaring Algorithm

When you are squaring a value, you can take advantage of the fact that
half the multiplications performed by the more general multiplication
algorithm (see 'mul.txt' for a description) are redundant when the
multiplicand equals the multiplier.

In particular, the modified algorithm is:

k = 0
for j <- 0 to (#a - 1)
  w = c[2*j] + (a[j] ^ 2);
  k = w div R

  for i <- j+1 to (#a - 1)
    w = (2 * a[j] * a[i]) + k + c[i+j]
    c[i+j] = w mod R
    k = w div R
  endfor
  c[i+j] = k;
  k = 0;
endfor

On the surface, this looks identical to the multiplication algorithm;
however, note the following differences:

  - precomputation of the leading term in the outer loop

  - i runs from j+1 instead of from zero

  - doubling of a[i] * a[j] in the inner product

Unfortunately, the construction of the inner product is such that we
need more than two digits to represent the inner product, in some
cases.  In a C implementation, this means that some gymnastics must be
performed in order to handle overflow, for which C has no direct
abstraction.  We do this by observing the following:

If we have multiplied a[i] and a[j], and the product is more than half
the maximum value expressible in two digits, then doubling this result
will overflow into a third digit.  If this occurs, we take note of the
overflow, and double it anyway -- C integer arithmetic ignores
overflow, so the two digits we get back should still be valid, modulo
the overflow.

Having doubled this value, we now have to add in the remainders and
the digits already computed by earlier steps.  If we did not overflow
in the previous step, we might still cause an overflow here.  That
will happen whenever the maximum value expressible in two digits, less
the amount we have to add, is greater than the result of the previous
step.  Thus, the overflow computation is:


  u = 0
  w = a[i] * a[j]

  if(w > (R - 1)/ 2)
    u = 1;

  w = w * 2
  v = c[i + j] + k

  if(u == 0 && (R - 1 - v) < w)
    u = 1

If there is an overflow, u will be 1, otherwise u will be 0.  The rest
of the parameters are the same as they are in the above description.

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$Id: square.txt,v 1.2 2005/02/02 22:28:22 gerv%gerv.net Exp $