A Convolve-And-MErge Approach for Exact Computations on High-Performance Reconfigurable Computers

Table 1

Computational complexity of arithmetic operations [18].

Operation	Input	Output	Algorithm	Complexity
Addition	Two n-digit numbers	One (𝑛+1)-digit number	Basecase/Schoolbook	𝑂(𝑛)
Subtraction	Two n-digit numbers	One (𝑛+1)-digit number	Basecase/Schoolbook	𝑂(𝑛)
Multiplication	Two n-digit numbers	One 2n-digit number	Basecase/Schoolbook	𝑂(𝑛^2)
			Karatsuba	𝑂(𝑛^1.585)
			3-way Toom-Cook	𝑂(𝑛^1.465)
			k-way Toom-Cook	𝑂(𝑛^1+𝜀),𝜀>0
			Mixed-level Toom-Cook	𝑂(𝑛(log𝑛)2^(2log𝑛))
			Schösnhage-Strassen	𝑂(𝑛(log𝑛)(loglog𝑛))
Note: The complexity of multiplication will be referred to as M(n) in the following
Division	Two n-digit numbers	One n-digit number	Basecase/Schoolbook	𝑂(𝑛^2)
			Newton’s method	𝑂(𝑀(𝑛))
			Goldschmidt	𝑂(𝑀(𝑛))
Square root	One n-digit number	One n-digit number	Newton’s method	𝑂(𝑀(𝑛))
			Goldschmidt	𝑂(𝑀(𝑛))
Polynomial evaluation	n fixed-size polynomial coefficients	One fixed size	Horner’s method	𝑂(𝑛)
			Direct evaluation	𝑂(𝑛)