Fast parallel polynomial division via reduction to triangular toeplitz matrix inversion and to polynomial inversion modulo a power

Abstract

It is known that division with a remainder of two polynomials of degree at most s can be performed over an arbitrary field F of constants using uniform arithmetic and Boolean circuits of depth O(log s log log s) and polynomial size. A new algorithm is presented that yields those bounds via reduction to triangular Toeplitz matrix inversion and to polynomial inversion modulo a power. (If|F| > (s−1) 2 or if P-uniform computation is allowed, then the depth can be reduced to O(log s).) This approach is new and makes the result conceptually simpler.