We show that circuit lower bound proofs based on the method of random restrictions yield non-trivial compression algorithms for ``easy'' Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an n-variate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2n) a circuit C (no restriction on the type of C) computing f so that the size of C is less than the trivial circuit size 2nn . We get non-trivial compression for functions computable by AC0 circuits, (de Morgan) formulas, and (read-once) branching programs of the size for which the lower bounds for the corresponding circuit class are known.

These compression algorithms rely on the structural characterizations of ``easy'' functions, which are useful both for proving circuit lower bounds and for designing ``meta-algorithms'' (such as Circuit-SAT). For (de Morgan) formulas, such structural characterization is provided by the ``shrinkage under random restrictions'' results [Sub61,Has98], strengthened to the ``high-probability'' version by [San10,IMZ12,KR12]. We give a new, simple proof of the ``high-probability'' version of the shrinkage result for (de Morgan) formulas, with improved parameters. We use this shrinkage result to get both compression and #SAT algorithms for (de Morgan) formulas of size about n2. We also use this shrinkage result to get an alternative proof of the recent result by Komargodski and Raz [KR12] of the average-case lower bound against small (de Morgan) formulas.

Finally, we show that the existence of any non-trivial compression algorithm for a circuit class CP/poly would imply the circuit lower bound that NEXP is not in C. This complements Williams's result [Wil10] that any non-trivial Circuit-SAT algorithm for a circuit class C would imply a superpolynomial lower bound against C for a language in NEXP