Implementation of the DKSS Algorithm for Multiplication of Large Numbers

Abstract

The Schönhage-Strassen algorithm (SSA) is the de-facto standard for multiplication of large integers. For N -bit numbers it has a time bound of O(N · log N · log log N ). De, Kurur, Saha and Saptharishi (DKSS) presented an asymptotically faster algorithm with a better time bound of N · log N · 2 O(log * N ) . For this paper, a simplified DKSS multiplication was implemented. Assuming a sensible upper limit on the input size, some required constants could be precomputed. This allowed to simplify the algorithm to save some complexity and run-time. Still, run-time is about 30 times larger than SSA, while memory requirements are about 2.3 times higher than SSA. A possible crossover point is estimated to be out of reach even if we utilized the whole universe for computer memory.

Fast Polynomial Multiplication over F260

“…However, no-one has yet demonstrated a practical implementation for sizes supported by current technology. The implementation of the modular variant proposed in [7] has even been discussed in detail in [28]: their conclusion is that the break-even point seems to be beyond astronomical sizes.…”

Parallel integer multiplication

“…Schönhage-Strassen's algorithm [1,2] multiplies two integers of size n in O(n • log n • log log n) and is to this day the best algorithm known for integers of reachable sizes. Asymptotically faster algorithms exist [3], however no computer is able to hold numbers big enough for those algorithms to outrun Schönhage-Strassen [4]. If M (n) is the cost of multiplying two integers of size n, many arithmetic operations can be computed in either O(M (n)) or O(M (n)•log n) such as the greatest common divisor [5], square root, and division [6].…”

Speeding up the Multiplication Algorithm for Large Integers

“…The second direction has been to reduce the cost of multiplying the data set by proposing an efficient strategy to multiply the ݉ numbers. Regarding the first research direction, many methods have been proposed to reduce the time complexity of multiplying two integers in both sequential [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] and parallel computation [24][25][26][27][28][29]. In the case of sequential computation, several techniques have been proposed such as the Naïve multiplication algorithm [1], Karatsuba's algorithm [1,15], the Toom-Cook multiplication algorithm [20], and a fast Fourier transform-based algorithm [18].…”