On the Size of Monotone Span Programs

Span programs provide a linear algebraic model of computation. Monotone span programs (MSP) correspond to linear secret sharing schemes. This paper studies the properties of monotone span programs related to their size. Using the results of van Dijk (connecting codes and MSPs) and a construction for a dual monotone span program proposed by Cramer and Fehr we prove a non-trivial upper bound for the size of monotone span programs. By combining the concept of critical families with the dual monotone span program construction of Cramer and Fehr we improve the known lower bound with a constant factor, showing that the lower bound for the size of monotone span programs should be approximately twice as large. Next we extend the result of Van Dijk showing that for any MSP there exists a dual MSP such that the corresponding codes are dual. Finally, we introduce the notions of redundant monotone span programs. Then we prove that a restricted non-redundant monotone span program exists if and only if a monotone dependency program exists.