On the multiplicative complexity of Boolean functions over the basis ∧,⊕,1
Theoretical Computer Science - Selected papers in honor of Manuel Blum
Tight bounds for the multiplicative complexity of symmetric functions
Theoretical Computer Science
Affine dispersers from subspace polynomials
Proceedings of the forty-first annual ACM symposium on Theory of computing
New upper bounds on the Boolean circuit complexity of symmetric functions
Information Processing Letters
A well-mixed function with circuit complexity 5n ± o(n): tightness of the Lachish-Raz-type bounds
TAMC'08 Proceedings of the 5th international conference on Theory and applications of models of computation
Circuit complexity and multiplicative complexity of Boolean functions
CiE'10 Proceedings of the Programs, proofs, process and 6th international conference on Computability in Europe
A 5n−o(n) lower bound on the circuit size over U2 of a linear Boolean function
CiE'12 Proceedings of the 8th Turing Centenary conference on Computability in Europe: how the world computes
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A Boolean function f: F2n → F2 is called an affine disperser of dimension d, if f is not constant on any affine subspace of F2n of dimension at least d. Recently Ben-Sasson and Kopparty gave an explicit construction of an affine disperser for sublinear d. The main motivation for studying such functions comes from extracting randomness from structured sources of imperfect randomness. In this paper, we show another application: we give a very simple proof of a 3n-o(n) lower bound on the circuit complexity (over the full binary basis) of affine dispersers for sublinear dimension. The same lower bound 3n-o(n) (but for a completely different function) was given by Blum in 1984 and is still the best known. The main technique is to substitute variables by linear functions. This way the function is restricted to an affine subspace of F2n. An affine disperser for sublinear dimension then guarantees that one can make n - o(n) such substitutions before the function degenerates. It remains to show that each such substitution eliminates at least 3 gates from a circuit.