Krylov methods rely on iterated matrix-vector products Akuj for an n \texttimes{} n matrix A and vectors u1, …, um. The space spanned by all iterates Akuj admits a particular basis — the maximal Krylov basis — which consists of iterates of the first vector u1, Au1, A2u1, …, until reaching linear dependency, then iterating similarly the subsequent vectors until a basis is obtained. Finding minimal polynomials and Frobenius normal forms is closely related to computing maximal Krylov bases. The fastest way to produce these bases was, until this paper, Keller-Gehrig’s 1985 algorithm whose complexity bound <?TeX $mathchoice{Oleft(n^omega log (n)right)}{O(n^omega log (n))}{O(n^omega log (n))}{O(n^omega log (n))}$?>Math 1 comes from repeated squarings of A and logarithmically many Gaussian eliminations. Here ω > 2 is a feasible exponent for matrix multiplication over the base field. We present an algorithm computing the maximal Krylov basis in <?TeX $mathchoice{Oleft(n^omega operatorname{loglog}(n)right)}{O(n^omega operatorname{loglog}(n))}{O(n^omega operatorname{loglog}(n))}{O(n^omega operatorname{loglog}(n))}$?>Math 2 field operations when <?TeX $m in mathchoice{Oleft(nright)}{O(n)}{O(n)}{O(n)}$?>Math 3, and even <?TeX $mathchoice{Oleft(n^omega right)}{O(n^omega)}{O(n^omega)}{O(n^omega)}$?>Math 4 as soon as <?TeX $min mathchoice{Oleft(n/log (n)^cright)}{O(n/log (n)^c)}{O(n/log (n)^c)}{O(n/log (n)^c)}$?>Math 5 for some fixed real c > 0. As a consequence, we show that the Frobenius normal form together with a transformation matrix can be computed deterministically in <?TeX $mathchoice{Oleft(n^omega (operatorname{loglog}(n))^2right)}{O(n^omega (operatorname{loglog}(n))^2)}{O(n^omega (operatorname{loglog}(n))^2)}{O(n^omega (operatorname{loglog}(n))^2)}$?>Math 6, and therefore matrix exponentiation Ak can be performed in the latter complexity if <?TeX $log (k) in mathchoice{Oleft(n^{omega -1-varepsilon }right)}{O(n^{omega -1-varepsilon })}{O(n^{omega -1-varepsilon })}{O(n^{omega -1-varepsilon })}$?>Math 7 for some fixed ε > 0. A key idea for these improvements is to rely on fast algorithms for m \texttimes{} m polynomial matrices of average degree n/m, involving high-order lifting and minimal kernel bases.