__
TR15-170 | 26th October 2015 17:10
__

#### Weighted gate elimination: Boolean dispersers for quadratic varieties imply improved circuit lower bounds

**Abstract:**
In this paper we motivate the study of Boolean dispersers for quadratic varieties by showing that an explicit construction of such objects gives improved circuit lower bounds. An $(n,k,s)$-quadratic disperser is a function on $n$ variables that is not constant on any subset of $\mathbb{F}_2^n$ of size at least $s$ that can be defined as the set of common roots of at most $k$ quadratic polynomials. We show that if a Boolean function $f$ is a $\left(n, 1.83n, 2^{g(n)}\right)$-quadratic disperser for any function $g(n)=o(n)$ then the circuit size of $f$ is at least $3.11n$. In order to prove this, we generalize the gate elimination method so that the induction works on the size of the variety rather than on the number of variables as in previously known proofs.