How to generate cryptographically strong sequences of pseudo-random bits
SIAM Journal on Computing
The complexity of Boolean functions
The complexity of Boolean functions
Small-bias probability spaces: efficient constructions and applications
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Learning decision trees using the Fourier spectrum
SIAM Journal on Computing
Journal of Computer and System Sciences
On lower bounds for read-k-times branching programs
Computational Complexity
P = BPP if E requires exponential circuits: derandomizing the XOR lemma
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
A read-once lower bound and a (1, +k)-hierarchy for branching programs
Theoretical Computer Science
Worst-Case Hardness Suffices for Derandomization: A New Method for Hardness-Randomness Trade-Offs
ICALP '97 Proceedings of the 24th International Colloquium on Automata, Languages and Programming
Time Space Tradeoffs (Getting Closer to the Barrier?)
ISAAC '93 Proceedings of the 4th International Symposium on Algorithms and Computation
Efficient Construction of Hitting Sets for Systems of Linear Functions
STACS '97 Proceedings of the 14th Annual Symposium on Theoretical Aspects of Computer Science
Lower Bounds for Deterministic and Nondeterministic Branching Programs
FCT '91 Proceedings of the 8th International Symposium on Fundamentals of Computation Theory
CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
Discrepancy sets and pseudorandom generators for combinatorial rectangles
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Simple construction of almost k-wise independent random variables
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
A Lower Bound Technique for Restricted Branching Programs and Applications
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
Almost k-wise independence and hard Boolean functions
Theoretical Computer Science - Latin American theoretical informatics
Lower Bounds for Syntactically Multilinear Algebraic Branching Programs
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
A well-mixed function with circuit complexity 5n: Tightness of the Lachish-Raz-type bounds
Theoretical Computer Science
Representation of graphs by OBDDs
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
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In several previous works the construction of a computationally hard function with respect to a certain class of algorithms or Boolean circuits has been used to derive small pseudo-random spaces. In this paper, we revert this connection by presenting two new direct relations between the efficient construction of pseudo-random (both two-sided and one-sided) sets for Boolean affine spaces and the explicit construction of Boolean functions having hard branching program complexity. In the case of 1-read branching programs (1-Br.Pr.), we show that the construction of non trivial (i.e. of cardinality 2o(n)) discrepancy sets (i.e. two-sided pseudo-random sets) for Boolean affine spaces of dimension greater than n/2 yield a set of explicit Boolean functions having very hard 1-Br.Pr. size. By combining the best known construction of Ɛ-biased sample spaces for linear tests and a simple "Reduction" Lemma, we derive the required discrepancy set and obtain a Boolean function in P having 1-Br.Pr. size not smaller than 2n-O(log2 n) and a Boolean function in DTIME(2O(log2 n)) having 1-Br.Pr. size not smaller than 2n-O(log n). The latter bound is optimal and both of them are exponential improvements over the best previously known lower bound that was 2n-3n1=2 [21]. As for non deterministic syntactic k-read branching programs (k-Br.Pr.), we introduce a new method to derive explicit, exponential lower bounds that involves the construction of hitting sets (one-sided pseudo-random sets) for affine spaces of dimension o(n/2). Using an appropriate "orthogonal" representation of small Boolean affine spaces, we efficiently construct these hitting sets thus obtaining an explicit Boolean function in P that has k-Br.Pr. size not smaller than 2n1-o(1) for any k = o(log n/log log n. This improves over the previous best known lower bounds given in [8,11, 17] for some range of k.