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The study of monotonicity and negation complexity for Boolean functions has been prevalent in complexity theory as well as in computational learning theory, but little attention has been given to it in the cryptographic context. Recently, Goldreich and Izsak (2012) have initiated a study of whether cryptographic primitives can be monotone, and showed that(More)
What we'd like: f 2 NP f / 2 P/poly such that. What we'd like: f 2 NP f / 2 P/poly such that. Explicit circuit lower bounds known for very few circuit classes. What we'd like: f 2 NP f / 2 P/poly such that. Explicit circuit lower bounds known for very few circuit classes. Even for low depth circuits of AND, OR, NOT and mod-m gates, a lower bound eluded us(More)
Different techniques have been used to prove several transference theorems of the form " non-trivial algorithms for a circuit class C yield circuit lower bounds against C ". In this survey we revisit many of these results. We discuss how circuit lower bounds can be obtained from deran-domization, compression, learning, and satisfiability algorithms. We also(More)
Fort now and Klivans proved the following relationship between efficient learning algorithms and circuit lower bounds: if a class of boolean circuits C contained in P/poly of Boolean is exactly learnable with membership and equivalence queries in polynomial-time, then EXP^NP is not contained in C (the class EXP^NP was subsequently improved to EXP by(More)
We consider C-compression games, a hybrid model between computational and communication complexity. A C-compression game for a function f : {0, 1} n → {0, 1} is a two-party communication game, where the first party Alice knows the entire input x but is restricted to use strategies computed by C-circuits, while the second party Bob initially has no(More)
Let U k,N denote the Boolean function which takes as input k strings of N bits each, representing k numbers a 2 N −1}, and outputs 1 if and only if a (1) +· · ·+a (k) ≥ 2 N. Let THR t,n denote a monotone unweighted threshold gate, i.e., the Boolean function which takes as input a single string x ∈ {0, 1} n and outputs 1 if and only if x 1 + · · · + x n ≥ t.(More)
We prove several results giving new and stronger connections between learning theory, circuit complexity and pseudorandomness. Let C be any typical class of Boolean circuits, and C[s(n)] denote n-variable C-circuits of size ≤ s(n). We show: Learning Speedups. If C[poly(n)] admits a randomized weak learning algorithm under the uniform distribution with(More)