Abstract
Motivated by an open problem recently suggested by Goldreich et al., we study truthful implementations of a random binary function supporting compound XOR queries over sub-cubes of the hypercube {0,1}n. We introduce a relaxed model of an implementation, which we call a stateful implementation, and show how to implement the desired specification in this model. The main technical construction is an algorithm for detecting linear dependencies between n dimensional hypercubes, viewed as characteristic vectors in \(\mathbb{F}_2^{\{0,1\}^n}\). Using coding theoretic techniques, we first exhibit a randomized algorithm for detecting such dependencies. We then show how a recent approach by Raz and Shpilka for polynomial identity testing in non-commutative models of computation can be applied to obtain a deterministic algorithm.
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Bogdanov, A., Wee, H. (2004). A Stateful Implementation of a Random Function Supporting Parity Queries over Hypercubes. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2004 2004. Lecture Notes in Computer Science, vol 3122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27821-4_27
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DOI: https://doi.org/10.1007/978-3-540-27821-4_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22894-3
Online ISBN: 978-3-540-27821-4
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