  We present a new algorithm for computing the characteristic polynomial of an
arbitrary endomorphism of a finite Drinfeld module using its associated
crystalline cohomology. Our approach takes inspiration from Kedlaya's p-adic
algorithm for computing the characteristic polynomial of the Frobenius
endomorphism on a hyperelliptic curve using Monsky-Washnitzer cohomology. The
method is specialized using a baby-step giant-step algorithm for the particular
case of the Frobenius endomorphism, and in this case we include a complexity
analysis that demonstrates asymptotic gains over previously existing approaches
