Circuit and Decision Tree Complexity of Some Number Theoretic Problems

We extend the area of applications of our methods to lower bounds on the circuit and decision tree complexity of Boolean functions related to some number-theoretic problems. In particular, we show that deciding whether a given integer is square-free and testing co-primality of two integers by unbounded fan-in circuits of bounded depth requires superpolynomial size, see [9, 38, 39, 40, 41, 460]. This method can be applied to other number-theoretic problems related to arithmetical properties of integers. Unfortunately this approach does not seem to apply to the complexity of the primality testing problem, for which an alternative approach has been developed in [9], which in fact implies stronger but less explicit results (which apply to square-freeness and co-primality testing as well).

Summary form only given. It is proved that if P not=NP, then there exits a set in NP that is polynomial-time bounded truth-table reducible to no sparse set. By using the technique proving this result, intractability of several number theoretic decision problems, i.e. decision problems defined naturally from number theoretic problems, is investigated. It is shown that for those number theoretic decision problems, if it is not in P, then it is polynomial-time bounded truth-table reducible to no sparse set.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We consider the minimal number of AND and OR gates in monotone circuits for quadratic boolean functions, i.e. disjunctions of length-2 monomials. The single level conjecture claims that monotone single level circuits, i.e. circuits which have only one level of AND gates, for quadratic functions are not much larger than arbitrary monotone circuits. In this paper we disprove the conjecture: there are quadratic functions in n variables whose monotone circuits have linear size whereas their monotone single level circuits require ).

We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in mathsf {Quasi}text {-}mathsf {NP} = mathsf {NTIME}[n^{(log n)^{O(1)}}] and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes mathcal { C}, by showing that mathcal { C} admits non-trivial satisfiability and/or # SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial # SAT algorithm for a circuit class {mathcal C}. Say that a symmetric Boolean function f(x1,…,xn) is sparse if it outputs 1 on O(1) values of {sum }_{i} x_{i}. We show that for every sparse f, and for all “typical” mathcal { C}, faster # SAT algorithms for mathcal { C} circuits imply lower bounds against the circuit class f circ mathcal { C}, which may be stronger than mathcal { C} itself. In particular: # SAT algorithms for nk-size mathcal { C}-circuits running in 2n/nk time (for all k) imply NEXP does not have (f circ mathcal { C})-circuits of polynomial size.# SAT algorithms for 2^{n^{{varepsilon }}}-size mathcal { C}-circuits running in 2^{n-n^{{varepsilon }}} time (for some ε > 0) imply Quasi-NP does not have (f circ mathcal { C})-circuits of polynomial size.Applying # SAT algorithms from the literature, one immediate corollary of our results is that Quasi-NP does not have EMAJ ∘ ACC0 ∘ THR circuits of polynomial size, where EMAJ is the “exact majority” function, improving previous lower bounds against ACC0 [Williams JACM’14] and ACC0 ∘THR [Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.

It is well known that among the three classes of the PRAM models, namely, CRCW, CREW, and EREW, the CRCW models are the weakest, in the sense that, they permit concurrent read/write by processors. Accordingly, algorithms on the CRCW model mainly concentrate on the core computations without much ado about data access. Consequently, this model, at least in principle, allows for the design of the fastest algorithm for a problem. It is intriguing to ask how fast prefixes can be computed on the CRCW models. Since CRCW models are equivalent to the unbounded fan-in circuits (refer to Chapter 2), the task of developing the fastest algorithms for the prefix problems is pursued in the context of the unbounded fan-in circuits. Recall from Chapter 2, that while the standard measures, such as, size and depth are still used to quantify the goodness of unbounded fanin circuits, the size of the circuit is measured by the total number of edges incident on all of its operation nodes, instead of by the number of operations nodes. It turns out that the size and depth of unbounded fanin circuits for computing prefixes, depends critically on the structure of the underlying semigroup from which the input elements are drawn. The principal result of this concluding Chapter may be stated as follows: There exists unbounded fan-in parallel prefix circuits of constant depth and polynomial size if, and only if, the underlying semigroup is group free. The proof of this result involves a very clever synthesis of a number of ideas drawn from different directions — structure of group free semigroups, their relations to a special class of regular sets, called non-counting regular sets, the relation of this latter class of regular sets to yet another class of regular sets defined by star-free regular expressions, and the design of a special class of finite state deterministic automata called RS machines that accept star-free regular expressions. In this context, it is convenient to define the notion of small circuits as the class of circuits with constant depth and polynomial size.

Limiting negations in bounded-depth circuits: An extension of Markov's theorem

Chapter 5 - Multiplication and Squaring

Most of the work done in cryptography in the last few years depend on the hardness of a few specific number theoretic problems, such as factoring, discrete log, etc. Since no one has so far been able to prove that these problems are genuinely hard, it is clearly of interest to find new candidates for hard problems. In this paper, we propose such a new candidate problem. namely the problem of predicting a sequence of consecutive Legendre (Jacobi) symbols modulo a prime (composite), when the starting point and possibly also the prime is unknown. Clearly, if this problem turns out to be hard, it can be used directly to construct a cryptographically strong pseudorandom bitgenerator. Its complexity seems to be unrelated to any of the well known number theoretical problems, whence it may be able to survive the discovery of fast factoring or discrete log algorithms. Although the randomness of Legendre sequences has part of the folklore in number theory at least since the thirties, they have apparently not been considered for use in cryptography before.

A number theoretic problem on super line graphs

The use of alternative foundations for constructing more secure and efficient cryptographic schemes is a topic worth exploring. In the case of proxy re-encryption, the vast majority of schemes are based on number theoretic problems such as the discrete logarithm. In this paper we present NTRUReEncrypt, a new bidirectional and multihop proxy re-encryption scheme based on NTRU, a widely known lattice-based cryptosystem. We provide two versions of our scheme: the first one is based on the conventional NTRU encryption scheme and, although it lacks a security proof, remains as efficient as its predecessor; the second one is based on a variant of NTRU proposed by Stehle and Steinfeld, which is proven CPA-secure under the hardness of the Ring-LWE problem. To the best of our knowledge, our proposals are the first proxy re-encryption schemes to be based on the NTRU primitive. In addition, we provide experimental results to show the efficiency of our proposal, as well as a comparison with previous proxy re-encryption schemes, which confirms that our first scheme outperforms the rest by an order of magnitude.

Optimal Depth, Very Small Size Circuits for Symmetrical Functions in AC0

The precise manipulation of neural activity is of great significance to reveal the function of complex neural circuits in brain, especially the mechanism of neurological disorders such as epilepsy, Parkinson's, and Alzheimer's disease. In the past decades, optogenetics has become a powerful tool to investigate the function of neural circuits due to its unique advantages of cell specificity and high spatiotemporal resolution. However, to date, most optogenetic manipulation has been conducted in an open‐loop fashion. The closed‐loop neural modulation is still an important task. Here, a closed‐loop optogenetic modulation system is developed for ultra‐fast, ‐accurate, and ‐stable manipulation of neural activities in behaving mice. Specifically, this system is mediated by a functional fiber probe with excellent low optical loss (0.052 dB cm−1 at 589 nm) and electrical impedance (5.40 MΩ µm−2 at 1 kHz). In vivo electrophysiology and behavioral experiments demonstrated that the system can be successfully applied for the identification and intervention of seizures in epilepsy mice with high efficiency. It is envision that the closed‐loop manipulation system has promising applications in brain science, neuroscience, and neuromodulation therapies.

Lower bounds on the area complexity of Boolean circuits

In this paper we initiate study of the computational power of adaptive and non-adaptive monotone decision trees - decision trees where each query is a monotone function on the input bits. In the most general setting, the monotone decision tree height (or size) can be viewed as a measure of non-monotonicity of a given Boolean function. We also study the restriction of the model by restricting (in terms of circuit complexity) the monotone functions that can be queried at each node. This naturally leads to complexity classes of the form \({\mathsf {DT}}(\textit{mon-}\mathcal {C})\) for any circuit complexity class \(\mathcal {C}\), where the height of the tree is \(\mathcal {O}(\log n)\), and the query functions can be computed by monotone circuits in class \(\mathcal {C}\). In the above context, we prove the following characterizations and bounds.

How are functional neural circuits formed during development? Despite recent advances in our understanding of the development of individual neurons, little is known about how complex circuits are assembled to generate specific behaviors. Here, we describe the ways in which Drosophila motor circuits serve as an excellent model system to tackle this problem. We first summarize what has been learned during the past decades on the connectivity and development of component neurons, in particular motor neurons and sensory feedback neurons. We then review recent progress in our understanding of the development of the circuits as well as studies that apply optogenetics and other innovative techniques to dissect the circuit diagram. New approaches using Drosophila as a model system are now making it possible to search for developmental rules that regulate the construction of neural circuits.