On the Shortest Linear Straight-Line Program for Computing Linear Forms

Authors:
Joan Boyar;Philip Matthews;René Peralta
Affiliations:
Dept. of Math. and Computer Science, University of Southern Denmark,;Dept. of Math. and Computer Science, University of Southern Denmark,;Computer Security Division, Information Technology Laboratory, NIST,
Venue:
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Year:
2008

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Abstract

We study the complexity of the Shortest Linear Program (SLP) problem, which is to minimize the number of linear operations necessary to compute a set of linear forms. SLP is shown to be NP-hard. Furthermore, a special case of the corresponding decision problem is shown to be MaxSNP-Complete.Algorithms producing cancellation-free straight-line programs, those in which there is never any cancellation of variables in GF(2), have been proposed for circuit minimization for various cryptographic applications. We show that such algorithms have approximation ratios of at least 3/2 and therefore cannot be expected to yield optimal solutions to non-trivial inputs.