Separating the power of monotone span programs over different fields

Monotone span programs represent a linear-algebraic model of computation. They are equivalent to linear secret sharing schemes and have various applications in cryptography and complexity. A fundamental question regarding them is how the choice of the field in which the algebraic operations are performed affects the power of the span program. In this paper we prove that the power of monotone span programs over finite fields of different characteristics is incomparable; we show a superpolynomial separation between any two fields with different characteristics, solving an open problem of Pudlák and Sgall [Algebraic models of computation and interpolation for algebraic proof systems, in Proof Complexity and Feasible Arithmetic, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 39, P. W. Beame and S. Buss, eds., AMS, Providence, RI, 1998, pp. 279-296]. Using this result we prove a superpolynomial lower bound for monotone span programs for a function in uniform-NC² (and therefore in P), solving an open problem of Babai, Gál, and Wigderson [Combinatorica, 19 (1999), pp. 301-319]. (All previous superpolynomial lower bounds for monotone span programs were for functions not known to be in P.) Finally, we show that quasi-linear secret sharing schemes, a generalization of linear secret sharing schemes introduced in Beimel and Ishai [On the power of nonlinear secret-sharing, in Proceedings of the 16th Annual IEEE Conference on Computational Complexity, 2001, pp. 188-202], are stronger than linear secret sharing schemes. In particular, this proves, without any assumptions, that nonlinear secret sharing schemes are more efficient than linear secret sharing schemes.