Separating the power of monotone span programs over different fields

Monotone span programs are 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 is how the choice of the field in which the algebraic operations are performed effects 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 super-polynomial separation between any two fields with different characteristics, answering an open problem of Pudlak and Sgall (1998). Using this result we prove a super-polynomial lower bound for monotone span programs for a function in uniform - 𝒩;𝒞;² (and therefore in 𝒫;), answering an open problem of Babai, Wigderson, and Gal (1999). Finally, we show that quasi-linear schemes, a generalization of linear secret sharing schemes introduced in Beimel and Ishai (2001), are stronger than linear secret sharing schemes. In particular, this proves, without any assumptions, that non-linear secret sharing schemes are more efficient than linear secret sharing schemes.