Division (digital) - Large Integer Methods

Large Integer Methods

Methods designed for hardware implementation generally do not scale to integers with thousands or millions of decimal digits; these frequently occur, for example, in modular reductions in cryptography. For these large integers, more efficient division algorithms transform the problem to use a small number of multiplications, which can then be done using an asymptotically efficient multiplication algorithm such as the Karatsuba algorithm, Toom–Cook multiplication or the Schönhage–Strassen algorithm. It results that the computational complexity of the division is of the same order (up a multiplicative constant) as that of the multiplication. Examples include reduction to multiplication by Newton's method as described above as well as the slightly faster Barrett reduction algorithm. Newton's method's is particularly efficient in scenarios where one must divide by the same divisor many times, since after the initial Newton inversion only one (truncated) multiplication is needed for each division.