Computational Complexity of Mathematical Operations - Arithmetic Functions

Arithmetic Functions

Operation Input Output Algorithm Complexity
Addition Two n-digit numbers One n+1-digit number Schoolbook addition with carry Θ(n)
Subtraction Two n-digit numbers One n+1-digit number Schoolbook subtraction with borrow Θ(n)
Multiplication Two n-digit numbers
 One 2n-digit number Schoolbook long multiplication O(n2)
Karatsuba algorithm O(n1.585)
3-way Toom–Cook multiplication O(n1.465)
k-way Toom–Cook multiplication O(nlog (2k − 1)/log k)
Mixed-level Toom–Cook (Knuth 4.3.3-T) O(n 2√2 log n log n)
Schönhage–Strassen algorithm O(n log n log log n)
Fürer's algorithm O(n log n 2log* n)
Division Two n-digit numbers One n-digit number Schoolbook long division O(n2)
Newton–Raphson division O(M(n))
Square root One n-digit number One n-digit number Newton's method O(M(n))
Modular exponentiation Two n-digit numbers and a k-bit exponent One n-digit number Repeated multiplication and reduction O(2kM(n))
Exponentiation by squaring O(k M(n))
Exponentiation with Montgomery reduction O(k M(n))

Schnorr and Stumpf conjectured that no fastest algorithm for multiplication exists.

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