Karatsuba algorithm

It is therefore asymptotically faster than the traditional algorithm, which performs

The Toom–Cook algorithm (1963) is a faster generalization of Karatsuba's method, and the Schönhage–Strassen algorithm (1971) is even faster, for sufficiently large n. The standard procedure for multiplication of two n-digit numbers requires a number of elementary operations proportional to

Within a week, Karatsuba, then a 23-year-old student, found an algorithm that multiplies two n-digit numbers in

Kolmogorov was very excited about the discovery; he communicated it at the next meeting of the seminar, which was then terminated.

Kolmogorov gave some lectures on the Karatsuba result at conferences all over the world (see, for example, "Proceedings of the International Congress of Mathematicians 1962", pp.

351–356, and also "6 Lectures delivered at the International Congress of Mathematicians in Stockholm, 1962") and published the method in 1962, in the Proceedings of the USSR Academy of Sciences.

The article had been written by Kolmogorov and contained two results on multiplication, Karatsuba's algorithm and a separate result by Yuri Ofman; it listed "A. Karatsuba and Yu.

Karatsuba only became aware of the paper when he received the reprints from the publisher.

[2] The basic principle of Karatsuba's algorithm is divide-and-conquer, using a formula that allows one to compute the product of two large numbers

This basic step is, in fact, a generalization of a similar complex multiplication algorithm, where the imaginary unit i is replaced by a power of the base.

where These formulae require four multiplications and were known to Charles Babbage.

can be computed in only three multiplications, at the cost of a few extra additions.

We use m right shifts for decomposing the input operands using the resulting base (Bm = 1000), as: Only three multiplications, which operate on smaller integers, are used to compute three partial results: We get the result by just adding these three partial results, shifted accordingly (and then taking carries into account by decomposing these three inputs in base 1000 as for the input operands): Note that the intermediate third multiplication operates on an input domain which is less than two times larger than for the two first multiplications, its output domain is less than four times larger, and base-1000 carries computed from the first two multiplications must be taken into account when computing these two subtractions.

If n is four or more, the three multiplications in Karatsuba's basic step involve operands with fewer than n digits.

Therefore, those products can be computed by recursive calls of the Karatsuba algorithm.

The recursion can be applied until the numbers are so small that they can (or must) be computed directly.

In a computer with a full 32-bit by 32-bit multiplier, for example, one could choose B = 231 and store each digit as a separate 32-bit binary word.

Then the sums x1 + x0 and y1 + y0 will not need an extra binary word for storing the carry-over digit (as in carry-save adder), and the Karatsuba recursion can be applied until the numbers to multiply are only one digit long.

Karatsuba's basic step works for any base B and any m, but the recursive algorithm is most efficient when m is equal to n/2, rounded up.

In particular, if n is 2k, for some integer k, and the recursion stops only when n is 1, then the number of single-digit multiplications is 3k, which is nc where c = log23.

Since one can extend any inputs with zero digits until their length is a power of two, it follows that the number of elementary multiplications, for any n, is at most

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 for some constants c and d. For this recurrence relation, the master theorem for divide-and-conquer recurrences gives the asymptotic bound

It follows that, for sufficiently large n, Karatsuba's algorithm will perform fewer shifts and single-digit additions than longhand multiplication, even though its basic step uses more additions and shifts than the straightforward formula.

For small values of n, however, the extra shift and add operations may make it run slower than the longhand method.

Here is the pseudocode for this algorithm, using numbers represented in base ten.

For the binary representation of integers, it suffices to replace everywhere 10 by 2.

This method may produce negative numbers, which require one extra bit to encode signedness, and would still require one extra bit for the multiplier.

However, one way to avoid this is to record the sign and then use the absolute value of

to perform an unsigned multiplication, after which the result may be negated when both signs originally differed.