Monotone span programs are a linear-algebraic model of computation which were introduced by Karchmer and Wigderson in 1993. They are known to be equivalent to linear secret sharing schemes, and have various applications in complexity theory and cryptography. Lower bounds for monotone span programs have been difficult to obtain because they use non-monotone operations to compute monotone functions; in fact, the best known lower bounds are quasipolynomial for a function in (nonmonotone) P. A fundamental open problem is to prove exponential lower bounds on monotone span program size for any explicit function.

We resolve this open problem by giving exponential lower bounds on monotone span program size for a function in monotone P. This also implies the first exponential lower bounds for linear secret sharing schemes. Our result is obtained by proving exponential lower bounds using Razborov's rank method, a measure that is strong enough to prove lower bounds for many monotone models. The best prior results on the rank measure were quasipolynomial lower bounds for a function in NP. As corollaries we obtain new proofs of exponential lower bounds for monotone formula size, monotone switching network size, and the first lower bounds for monotone comparator circuit size for a function in monotone P. We also obtain new polynomial degree lower bounds for Nullstellensatz refutations using an interpolation theorem of Pudlak and Sgall. Finally, we obtain quasipolynomial lower bounds on the rank measure for the st-connectivity function, implying tight bounds for st-connectivity in all of the computational models mentioned above.