Faster polynomial multiplication via multipoint Kronecker substitution

نویسنده

David Harvey

چکیده

We give several new algorithms for dense polynomial multiplication based on the Kronecker substitution method. For moderately sized input polynomials, the new algorithms improve on the performance of the standard Kronecker substitution by a sizeable constant, both in theory and in empirical tests.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Simultaneous Modular Reduction and Kronecker Substitution for Small Finite Fields

We present algorithms to perform modular polynomial multiplication or modular dot product efficiently in a single machine word. We pack polynomials into integers and perform several modular operations with machine integer or floating point arithmetic. The modular polynomials are converted into integers using Kronecker substitution (evaluation at a sufficiently large integer). With some control ...

متن کامل

Complexity of finite field arithmetic

For integers, the parameter n is the bit-length, and for the finite field Fq we let n = log q. In the case of polynomial root-finding, d is the degree of the polynomial and we list bounds on the expected running time since these operations are most efficiently implemented using probabilistic algorithms. In Lecture 3 we addressed the cost of addition and subtraction in both Z and Fq, and the cos...

متن کامل

Multiplication of sparse Laurent polynomials and Poisson series on modern hardware architectures

In this paper we present two algorithms for the multiplication of sparse Laurent polynomials and Poisson series (the latter being algebraic structures commonly arising in Celestial Mechanics from the application of perturbation theories). Both algorithms first employ the Kronecker substitution technique to reduce multivariate multiplication to univariate multiplication, and then use the schoolb...

متن کامل

4.1 Euclidean Division

We saw in Lecture 3 how to efficiently multiply integers, and, using Kronecker substitution, how to efficiently multiply polynomials with integer coefficients. This gives us what we need to multiply elements in finite fields, provided that we have a way to reduce the result to our standard representations of Fp ' Z/pZ and Fq ' Fp[x]/(f), using integers in [0, p− 1] and polynomials of degree les...

متن کامل