On the size of monotone span programs

Authors:
Ventzislav Nikov;Svetla Nikova;Bart Preneel
Affiliations:
Department of Mathematics and Computing Science, Eindhoven University of Technology, Eindhoven, The Netherlands;Department Electrical Engineering, ESAT/COSIC, Katholieke Universiteit Leuven, Heverlee-Leuven, Belgium;Department Electrical Engineering, ESAT/COSIC, Katholieke Universiteit Leuven, Heverlee-Leuven, Belgium
Venue:
SCN'04 Proceedings of the 4th international conference on Security in Communication Networks
Year:
2004

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Abstract

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. Finally, we extend the result of van Dijk showing that for any MSP there exists a dual MSP such that the corresponding codes are dual.