Prime Numbers

9.7 Exercises 519
(3) Write out complete algorithms for addition and subtraction of integers in
base B, with signs arbitrary.
9.3. Assume that each of two nonnegative integers x, y is given in balanced
base-B representation. Give an explicit algorithm for calculating the sum
x + y, digit by digit, but always staying entirely within the balanced base-B
representation for the sum. Then write out a such a self-consistent multiply
algorithm for balanced representations.
9.4. It is known to children that multiplication can be effected via addition
alone, as in 3 · 5 = 5 + 5 + 5. This simple notion can actually have
practical import in some scenarios (actually, for some machines, especially
older machines where word multiply is especially costly), as seen in the
following tasks, where we study how to use storage tricks to reduce the amount
of calculation during a large-integer multiply. Consider the multiplication of
D-digit, base-(B =2 b ) integers of size 2 n ,sothatn ≈ bD. For the tasks below,
define a “word” operation (word multiply or word add) as one involving two
size-B operands (each having b bits).
(1) Argue first that standard grammar-school multiply, whereby one constructs
via word multiplies a parallelogram and then adds up the columns
via word adds, requires O(D 2 ) word multiplies and O(D 2 ) word adds.
(2) Noting that there can be at most B possible rows of the parallelogram,
argue that all possible rows can be precomputed in such a way that the
full multiply requires O(BD) word multiplies and O(D 2 ) word adds.
(3) Now argue that the precomputation of all possible rows of the parallelogram
can be done with successive additions and no multiplies of any kind,
so that the overall multiply can be done in O(D 2 + BD) word adds.
(4) Argue that the grammar-school paradigm of task (1) above can be done
with O(n) bits of temporary memory. What, then, are the respective
memory requirements for tasks (2), (3)?
If one desires to create an example program, here is a possible task: Express
large integers in base B = 256 = 2 8 and implement via machine task (2)
above, using a 256-integer precomputed lookup table of possible rows to create
the usual parallelogram. Such a scheme may well be slower than other largeinteger
methods, but as we have intimated, a machine with especially slow
word multiply can benefit from these ideas.
9.5. Write out an explicit algorithm (or an actual program) that uses the
wn relation (9.3) to effect multiple-precision squaring in about half a multipleprecision
multiply time. Note that you do not need to subtract out the term
δn explicitly, if you elect instead to modify slightly the i sum. The basic point
is that the grammar-school rhombus is effectively cut (about) in half. This
exercise is not as trivial as it may sound—there are precision considerations
attendant on the possibility of huge column sums.