Composing Power Series Over a Finite Ring in Essentially Linear Time

Applications. The problem of computing f(g), under the restrictions deg f < n and g(0) = 0, is known as order-n power series composition. Here is the point: given power series u and v over R, with v(0) = 0, define f = u(z) mod z and g = v(x) mod x; then f(g) = u(v(x)) mod x. Power series composition is the bottleneck in reversion and iteration of power series. See Brent and Kung (1978) and Knuth (1981, Section 4.7).