Polynomial multiplication

The convolution of coefficient sequences $a_i,b_i$ has coefficients $c_n={\sum }_{i+j=n}{a}_i{b}_j$. These are exactly the coefficients of the product of the corresponding polynomials.

Carryless multiplication treats binary digits as coefficients of a polynomial over the field with two elements. Products are formed by polynomial multiplication, with exclusive or combining equal-degree terms instead of carrying to the next binary position.

Kronecker substitution encodes a polynomial with bounded nonnegative integer coefficients as one large integer by evaluating it at a sufficiently large base. Integer multiplication then gives the polynomial product, provided the base exceeds every output coefficient so carries cannot mix coefficients.