A lower bound for polynomial multiplication

Optimal Complexity Lower Bound for Polynomial Multiplication

We prove lower bounds of order n logn for both the problem of multiplying polynomials of degree n, and of dividing polynomials with remainder, in the model of bounded coefficient arithmetic circuits over the complex numbers. These lower bounds are optimal up to order of magnitude. The proof uses a recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix multiplication. It reduces the li...

A Lower Bound for Matrix Multiplication

We prove that computing the product of two 11 X 11 matrices over the binary field requires at least 2.5 11:1 0 (11:1) multiplications.

A Lower Bound on the Complexity of Polynomial Multiplication Over Finite Fields

A Depth-Five Lower Bound for Iterated Matrix Multiplication

We prove that certain instances of the iterated matrix multiplication (IMM) family of polynomials with N variables and degree n require NΩ( √ n) gates when expressed as a homogeneous depth-five ΣΠΣΠΣ arithmetic circuit with the bottom fan-in bounded by N1/2−ε. By a depth-reduction result of Tavenas, this size lower bound is optimal and can be achieved by the weaker class of homogeneous depth-fo...

Strassen’s lower bound for polynomial evaluation and Bezout’s theorem

Strassen’s lower bound for polynomial evaluation and Bezout’s theorem Recall Strassen’s algorithm from the previous lecture: Given: (a0, . . . , an−1), (x1, . . . , xn) ∈ K, and polynomial p(x) = ∑n−1 i=0 aix i Task: find (z1, . . . , zn), zi = p(xi) How many steps do we need to accomplish this task? Using the Fast Fourier Transform (FFT) we need O(n log n) steps. Strassen was interested whethe...