Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness and randomness extraction. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes

Pseudorandom Functions in Almost Constant Depth from Low-Noise LPN

Pseudorandom functions (PRFs) play a central role in symmetric cryptography. While in principle they can be built from any one-way functions by going through the generic HILL (SICOMP 1999) and GGM (JACM 1986) transforms, some of these steps are inherently sequential and far from practical. Naor, Reingold (FOCS 1997) and Rosen (SICOMP 2002) gave parallelizable constructions of PRFs in NC$^2$ and TC$^0$ based on concrete number-theoretic assumptions such as DDH, RSA, and factoring. Banerjee, Peikert, and Rosen (Eurocrypt 2012) constructed relatively more efficient PRFs in NC$^1$ and TC$^0$ based on ``learning with errors\u27\u27 (LWE) for certain range of parameters. It remains an open problem whether parallelizable PRFs can be based on the ``learning parity with noise\u27\u27 (LPN) problem for both theoretical interests and efficiency reasons (as the many modular multiplications and additions in LWE would then be simplified to AND and XOR operations under LPN). In this paper, we give more efficient and parallelizable constructions of randomized PRFs from LPN under noise rate $n^{-c}$ (for any constant 0<c<1) and they can be implemented with a family of polynomial-size circuits with unbounded fan-in AND, OR and XOR gates of depth $\omega(1)$, where $\omega(1)$ can be any small super-constant (e.g., $\log\log\log{n}$ or even less). Our work complements the lower bound results by Razborov and Rudich (STOC 1994) that PRFs of beyond quasi-polynomial security are not contained in AC$^0$(MOD$_2$), i.e., the class of polynomial-size, constant-depth circuit families with unbounded fan-in AND, OR, and XOR gates. Furthermore, our constructions are security-lifting by exploiting the redundancy of low-noise LPN. We show that in addition to parallelizability (in almost constant depth) the PRF enjoys either of (or any tradeoff between) the following: (1) A PRF on a weak key of sublinear entropy (or equivalently, a uniform key that leaks any $(1 - o(1))$-fraction) has comparable security to the underlying LPN on a linear size secret. (2) A PRF with key length $\lambda$ can have security up to $2^{O(\lambda/\log\lambda)}$, which goes much beyond the security level of the underlying low-noise LPN. where adversary makes up to certain super-polynomial amount of queries

Algorithms and Lower Bounds in Circuit Complexity

Computational complexity theory aims to understand what problems can be efficiently solved by computation. This thesis studies computational complexity in the model of Boolean circuits. Boolean circuits provide a basic mathematical model for computation and play a central role in complexity theory, with important applications in separations of complexity classes, algorithm design, and pseudorandom constructions. In this thesis, we investigate various types of circuit models such as threshold circuits, Boolean formulas, and their extensions, focusing on obtaining complexity-theoretic lower bounds and algorithmic upper bounds for these circuits. (1) Algorithms and lower bounds for generalized threshold circuits: We extend the study of linear threshold circuits, circuits with gates computing linear threshold functions, to the more powerful model of polynomial threshold circuits where the gates can compute polynomial threshold functions. We obtain hardness and meta-algorithmic results for this circuit model, including strong average-case lower bounds, satisfiability algorithms, and derandomization algorithms for constant-depth polynomial threshold circuits with super-linear wire complexity. (2) Algorithms and lower bounds for enhanced formulas: We investigate the model of Boolean formulas whose leaf gates can compute complex functions. In particular, we study De Morgan formulas whose leaf gates are functions with "low communication complexity". Such gates can capture a broad class of functions including symmetric functions and polynomial threshold functions. We obtain new and improved results in terms of lower bounds and meta-algorithms (satisfiability, derandomization, and learning) for such enhanced formulas. (3) Circuit lower bounds for MCSP: We study circuit lower bounds for the Minimum Circuit Size Problem (MCSP), the fundamental problem of deciding whether a given function (in the form of a truth table) can be computed by small circuits. We get new and improved lower bounds for MCSP that nearly match the best-known lower bounds against several well-studied circuit models such as Boolean formulas and constant-depth circuits

Near-Optimal Secret Sharing and Error Correcting Codes in AC0

We study the question of minimizing the computational complexity of (robust) secret sharing schemes and error correcting codes. In standard instances of these objects, both encoding and decoding involve linear algebra, and thus cannot be implemented in the class AC0. The feasibility of non-trivial secret sharing schemes in AC0 was recently shown by Bogdanov et al. (Crypto 2016) and that of (locally) decoding errors in AC0 by Goldwasser et al. (STOC 2007). In this paper, we show that by allowing some slight relaxation such as a small error probability, we can construct much better secret sharing schemes and error correcting codes in the class AC0. In some cases, our parameters are close to optimal and would be impossible to achieve without the relaxation. Our results significantly improve previous constructions in various parameters. Our constructions combine several ingredients in pseudorandomness and combinatorics in an innovative way. Specifically, we develop a general technique to simultaneously amplify security threshold and reduce alphabet size, using a two-level concatenation of protocols together with a random permutation. We demonstrate the broader usefulness of this technique by applying it in the context of a variant of secure broadcast