  The Todd polynomials $td_k=td_k(b_1,b_2,\dots,b_m)$ are defined by their
generating functions $$\sum_{k\ge 0} td_k s^k = \prod_{i=1}^m \frac{b_i
s}{e^{b_i s}-1}.$$ It appears as a basic block in Todd class of a toric
variety, which is important in the theory of lattice polytopes and in number
theory. We find generalized Todd polynomials arise naturally in MacMahon's
partition analysis, especially in Ehrhart series computation. We give fast
evaluation of generalized Todd polynomials for numerical $b_i$'s. In order to
do so, we develop fast operations in the quotient ring $\mathbb{Z}_p[[s]]$
modulo $s^{d+1}$ for large prime $p$. As applications, i) we recompute the
Ehrhart series of magic squares of order 6, which was first solved by the first
named author. The running time is reduced from $70$ days to about $1$ day; ii)
we give a polynomial time algorithm for Integer Linear Programming when the
dimension is fixed, with a good performance.
