We give an explicit pseudorandom generator (PRG) for constant-depth read-once formulas over the basis $\{\wedge, \vee, \neg\}$ with unbounded fan-in. The seed length of our PRG is $\widetilde{O}(\log(n/\varepsilon))$. Previously, PRGs with near-optimal seed length were known only for the depth-2 case (Gopalan et al. FOCS '12). For a constant depth ... more >>>

There are only a few known general approaches for constructing explicit pseudorandom generators (PRGs). The ``iterated restrictions'' approach, pioneered by Ajtai and Wigderson [AW89], has provided PRGs with seed length $\mathrm{polylog} n$ or even $\tilde{O}(\log n)$ for several restricted models of computation. Can this approach ever achieve the optimal seed ... more >>>

In this work, we initiate the study of the Minimum Circuit Size Problem (MCSP) in the quantum setting. MCSP is a problem to compute the circuit complexity of Boolean functions. It is a fascinating problem in complexity theory---its hardness is mysterious, and a better understanding of its hardness can have ... more >>>

We give PRG for depth-$d$, size-$m$ $\mathrm{AC}^0$ circuits with seed length $O(\log^{d-1}(m)\log(m/\varepsilon)\log\log(m))$. Our PRG improves on previous work [TX13, ST19, Kel21] from various aspects. It has optimal dependence on $\frac{1}{\varepsilon}$ and is only one “$\log\log(m)$” away from the lower bound barrier. For the case of $d=2$, the seed length tightly ... more >>>

A secret-sharing scheme enables a dealer to share a secret $s$ among $n$ parties such that only authorized subsets of parties, specified by a monotone access structure $f:\{0,1\}^n\to\{0,1\}$, can reconstruct $s$ from their shares. Other subsets of parties learn nothing about $s$.

The question of minimizing the (largest) share size ... more >>>

We provide compelling evidence for the potential of hardness-vs.-randomness approaches to make progress on the long-standing problem of derandomizing space-bounded computation.

Our first contribution is a derandomization of bounded-space machines from hardness assumptions for classes of uniform deterministic algorithms, for which strong (but non-matching) lower bounds can be unconditionally proved. ... more >>>

Trevisan and Vadhan (FOCS 2000) introduced the notion of (seedless) extractors for samplable distributions as a possible solution to the problem of extracting random keys for cryptographic protocols from weak sources of randomness.
They showed that under a very strong complexity theoretic assumption, there exists a constant $\alpha>0$ such that ... more >>>

The hardness vs. randomness paradigm converts a function $f \colon \{0,1\}^n \rightarrow \{0,1\}$ that is hard for circuits of size $s$ into a pseudorandom generator (PRG) $G \colon \{0,1\}^d \to \{0,1\}^{s'}$ that fools circuits of size $s' = s'(s)$. In the application for derandomization, such as proofs of $\mathbf{BPP} = ... more >>>

We construct improved pseudorandom generators for read-$k$ oblivious branching programs with a known reading sequence.
For width-$w$ branching programs over $n$ variables, and designated error $\varepsilon$, our generator has seed length
$$\mathcal{O}\left(n^{1-\frac{1}{2k-1}}\log n\left(k\log w+\log\frac{n}{\varepsilon}\right)\right).$$
This improves upon the previous state-of-the-art due to Gurjar and Volk (ACM ToCT 2020), that has ... more >>>

We study cryptographic primitives that are both locally computable (i.e., in $\mathrm{NC}^0$) and exponentially secure. For pseudorandom generators (PRGs) and universal one-way hash functions (UOWHFs), we further require linear stretch and linear compression, respectively, which is essentially the best one can hope for in this setting. Such primitives simultaneously achieve ... more >>>

We construct explicit hitting set generators for $\mathbb{F}_2$-polynomials in $n$ variables of degree at most $d$ with an almost optimal density, and seed length $O(d \log n+d^2)$, which is optimal for $d = O(\log n)$.

Second, we address large-field hitting set generators. The influential state-of-the-art construction by Guruswami and Xing ... more >>>