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# Pseudorandom Functions in Tc0 and Cryptographic Limitations to Proving Lower Bounds

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## Abstract

This paper investigates which complexity classes inside NCcan contain pseudorandom function generators (PRFGs). Under the Decisional Diffie-Hellman assumption (a common cryptographic assumption) $$\textit{TC}^{0}$$4 contains PRFGs. No lower complexity classes with this property are currently known. On the other hand, we use effective lower bound arguments to show that some complexity classes cannot contain PRFGs. This provides evidence for the following conjecture: Any effective lower bound argument for a complexity class can be turned into an efficient distinguishing algorithm which proves that this class cannot

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... The moral is that, in order to prove stronger circuit lower bounds, one must avoid the techniques used in proofs that entail such efficient algorithms. The argument applies even to low-level complexity classes such as TC 0 [NR04, KL01,MV12], so any major progress in the future depends on proving un-Natural lower bounds. How should we proceed? ...
... Let C, D be appropriate circuit classes. Roughly speaking, the key lesson of Natural Proofs [RR97,NR04,KL01] is that, if there are D-natural properties useful against C, then there are no pseudorandom functions (PRFs) computable in C that fool D circuits; namely, there is a statistical test T computable in D such that, for every function f ∈ C (armed with an n-bit initial random seed), the test T with query access to f can distinguish f from a uniform random function. Now, if we have a PRF computable in C that can fool D circuits, this PRF can be used to obtain C seeds for randomized D circuits with one-sided error. ...
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We study connections between Natural Proofs, derandomization, and the problem of proving weak circuit lower bounds such as 'NEXP is not contained in TC^0' which are still wide open. Natural Proofs have three properties: they are constructive (an efficient algorithm ALG is embedded in them), have largeness (ALG accepts a large fraction of strings), and are useful (ALG rejects all strings which are truth tables of small circuits). Strong circuit lower bounds that are "naturalizing" would contradict present cryptographic understanding, yet the vast majority of known circuit lower bound proofs are naturalizing. So it is imperative to understand how to pursue un-Natural Proofs. Some heuristic arguments say constructivity should be circumventable. Largeness is inherent in many proof techniques, and it is probably our presently weak techniques that yield constructivity. We prove: * Constructivity is unavoidable, even for NEXP lower bounds. Informally, we prove for all "typical" non-uniform circuit classes C, NEXP is not contained in C if and only if there exists a constructive property that is nontrivially useful against C-circuits. * There are no P-natural properties useful against C if and only if randomized exponential time can be "derandomized" using truth tables of circuits from C as random seeds. Therefore the task of proving there are no P-natural properties is inherently a derandomization problem, weaker than but implied by the existence of strong pseudorandom functions. These characterizations are applied to yield several new results. The two main applications are that NEXP \cap coNEXP does not have n^{log n} size ACC circuits, and a mild derandomization result for RP.
... • PRFs in TC 0 4 based on the decisional Diffie-Hellman (DDH) assumption [KL01] (improving on [NR97]), yielding hardness for depth-6 ReLU networks We can now complete the proof of Theorem 6.1 by using Lemma 3.6 and Theorem 3.3: ...
Preprint
We give exponential statistical query (SQ) lower bounds for learning two-hidden-layer ReLU networks with respect to Gaussian inputs in the standard (noise-free) model. No general SQ lower bounds were known for learning ReLU networks of any depth in this setting: previous SQ lower bounds held only for adversarial noise models (agnostic learning) or restricted models such as correlational SQ. Prior work hinted at the impossibility of our result: Vempala and Wilmes showed that general SQ lower bounds cannot apply to any real-valued family of functions that satisfies a simple non-degeneracy condition. To circumvent their result, we refine a lifting procedure due to Daniely and Vardi that reduces Boolean PAC learning problems to Gaussian ones. We show how to extend their technique to other learning models and, in many well-studied cases, obtain a more efficient reduction. As such, we also prove new cryptographic hardness results for PAC learning two-hidden-layer ReLU networks, as well as new lower bounds for learning constant-depth ReLU networks from membership queries.
... This problem requires, for some integer k, a function that cannot be computed by a threshold circuit of width poly(d) and depth k, but can be computed 1 by a threshold circuit of width poly(d) and depth k ′ > k. Naor and Reingold [2004] and Krause and Lucks [2001] showed a candidate pseudorandom functions family computable by threshold circuits of depth 4, width poly(d), and poly(d)-bounded weights. By Razborov and Rudich [1997], it implies that for every k ′ > k ≥ 4, there is a natural-proof barrier for showing depth separation between threshold circuits of depth k and depth k ′ . ...
Preprint
In studying the expressiveness of neural networks, an important question is whether there are functions which can only be approximated by sufficiently deep networks, assuming their size is bounded. However, for constant depths, existing results are limited to depths $2$ and $3$, and achieving results for higher depths has been an important open question. In this paper, we focus on feedforward ReLU networks, and prove fundamental barriers to proving such results beyond depths $4$, by reduction to open problems and natural-proof barriers in circuit complexity. To show this, we study a seemingly unrelated problem of independent interest: Namely, whether there are polynomially-bounded functions which require super-polynomial weights in order to approximate with constant-depth neural networks. We provide a negative and constructive answer to that question, by showing that if a function can be approximated by a polynomially-sized, constant depth $k$ network with arbitrarily large weights, it can also be approximated by a polynomially-sized, depth $3k+3$ network, whose weights are polynomially bounded.
... The existence of pseudorandom functions follows from the existence of one-way functions ( [HILL99,GGM86]) which is essentially the weakest interesting cryptographic assumption. There are even candidate constructions of pseudorandom functions computable by polynomial-size constant-depth threshold circuits (TC 0 ) as given by Naor and Reingold [NR97], whose security rests on the intractability of discrete-log and factoring-type assumptions (see also Krause and Lucks [KL01]). As such, it is widely-believed that there are pseudorandom functions, even ones computationally indistinguishable from random except to adversaries running in exp(λ Ω(1) )-time. ...
Article
We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike the boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding whether algebraic circuits are expressive enough to support "cryptography" secure against algebraic circuits. Following a similar result of Williams in the boolean setting, we show that the existence of an algebraic natural proofs barrier is equivalent to the existence of succinct derandomization of the polynomial identity testing problem. That is, whether the coefficient vectors of polylog(N)-degree polylog(N)-size circuits is a hitting set for the class of poly(N)-degree poly(N)-size circuits. Further, we give an explicit universal construction showing that if such a succinct hitting set exists, then our universal construction suffices. Further, we assess the existing literature constructing hitting sets for restricted classes of algebraic circuits and observe that none of them are succinct as given. Yet, we show how to modify some of these constructions to obtain succinct hitting sets. This constitutes the first evidence supporting the existence of an algebraic natural proofs barrier. Our framework is similar to the Geometric Complexity Theory (GCT) program of Mulmuley and Sohoni, except that here we emphasize constructiveness of the proofs while the GCT program emphasizes symmetry. Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits.
... Theorem 1 implies that certain simple devices (namely, McCulloch-Pitts dynamical systems) cannot generate pseudorandomness. In the opposite direction, it has been proved that certain simple devices can generate pseudorandomness: examples can be found in [24], [19], [27], [26], [3]. Many examples of generators that appear random to observers with restricted computational powers are known. ...
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In a pioneering classic, Warren McCulloch and Walter Pitts proposed a model of the central nervous system. Motivated by EEG recordings of normal brain activity, Chvátal and Goldsmith asked whether these dynamical systems can be engineered to produce trajectories that are irregular, disorderly, and apparently unpredictable. We show that they cannot build weak pseudorandom functions.
... However, if some number-theoretic problems are exponentially hard on average (an assumption believed to be true by many researchers), then there are pseudorandom functions in circuit classes as small as TC 0 4 (Naor and Reingold [NR04], Krause and Lucks [KL01]). As a consequence, such proofs (dubbed natural proofs in [RR97]) are not expected to prove separations for more expressive circuit classes. ...
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Different techniques have been used to prove several transference theorems of the form "nontrivial algorithms for a circuit class C yield circuit lower bounds against C". In this survey we revisit many of these results. We discuss how circuit lower bounds can be obtained from derandomization, compression, learning, and satisfiability algorithms. We also cover the connection between circuit lower bounds and useful properties, a notion that turns out to be fundamental in the context of these transference theorems. Along the way, we obtain a few new results, simplify several proofs, and show connections involving different frameworks. We hope that our presentation will serve as a self-contained introduction for those interested in pursuing research in this area.
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We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures nearly all lower bound techniques known. However, unlike the boolean setting, there has been no concrete evidence demonstrating that this is a barrier to obtaining super-polynomial lower bounds for general algebraic circuits, as there is little understanding whether algebraic circuits are expressive enough to support "cryptography" secure against algebraic circuits. Following a similar result of Williams in the boolean setting, we show that the existence of an algebraic natural proofs barrier is equivalent to the existence of succinct derandomization of the polynomial identity testing problem. That is, whether the coefficient vectors of polylog(N)-degree polylog(N)-size circuits is a hitting set for the class of poly(N)-degree poly(N)-size circuits. Further, we give an explicit universal construction showing that if such a succinct hitting set exists, then our universal construction suffices. Further, we assess the existing literature constructing hitting sets for restricted classes of algebraic circuits and observe that none of them are succinct as given. Yet, we show how to modify some of these constructions to obtain succinct hitting sets. This constitutes the first evidence supporting the existence of an algebraic natural proofs barrier. Our framework is similar to the Geometric Complexity Theory (GCT) program of Mulmuley and Sohoni, except that here we emphasize constructiveness of the proofs while the GCT program emphasizes symmetry. Nevertheless, our succinct hitting sets have relevance to the GCT program as they imply lower bounds for the complexity of the defining equations of polynomials computed by small circuits.
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We prove that a single threshold gate with arbitrary weights can be simulated by an explicit polynomial-size depth 2 majority circuit. In general we show that a depth d threshold circuit can be simulated uniformly by a majority circuit of depth d + 1. Goldmann, Hastad, and Razborov showed in [10] that a non-uniform simulation exists. Our construction answers two open questions posed in [10]: we give an explicit construction whereas [10] uses a randomized existence argument, and we show that such a simulation is possible even if the depth d grows with the number of variables n (the simulation in [10] gives polynomial-size circuits only when d is constant). 1 A preliminary version of this paper appeared in Proc. 25th ACM STOC (1993), pp. 551--560. 2 Laboratory for Computer Science, MIT, Cambridge MA 02139, Email: migo@theory.lcs.mit.edu. This author 's work was done at Royal Institute of Technology in Stockholm, and while visiting the University of Bonn 3 Department of Com...
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In this paper, we prove the intractability of learning several classes of Boolean functions in the distribution-free model (also called the Probably Approximately Correct or PAC model) of learning from examples. These results are representation independent , in that they hold regardless of the syntactic form in which the learner chooses to represent its hypotheses. Our methods reduce the problems of cracking a number of well-known public-key cryptosystems to the learning problems. We prove that a polynomial-time learning algorithm for Boolean formulae, deterministic finite automata or constant-depth threshold circuits would have dramatic consequences for cryptography and number theory. In particular, such an algorithm could be used to break the RSA cryptosystem, factor Blum integers (composite numbers equivalent to 3 modulo 4), and detect quadratic residues. The results hold even if the learning algorithm is only required to obtain a slight advantage in prediction over random guessing. The techniques used demonstrate an interesting duality between learning and cryptography. We also apply our results to obtain strong intractability results for approximating a generalization of graph coloring.
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Luby and Rackoff [27] showed a method for constructing a pseudo-random permutation from a pseudo-random function. The method is based on composing four (or three for weakened security) so called Feistel permutations, each of which requires the evaluation of a pseudo-random function. We reduce somewhat the complexity of the construction and simplify its proof of security by showing that two Feistel permutations are sufficient together with initial and final pair-wise independent permutations. The revised construction and proof provide a framework in which similar constructions may be brought up and their security can be easily proved. We demonstrate this by presenting some additional adjustments of the construction that achieve the following: ffl Reduce the success probability of the adversary. ffl Provide a construction of pseudo-random permutations with large input size using pseudorandom functions with small input size. Incumbent of the Morris and Rose Goldman Career Development C...
The Probabilistic Method On the applications of multiplicity automata in learning
• N Alon
• J Spencer
• P Erd˝
N. Alon, J. Spencer & P. Erd˝ os (1992). The Probabilistic Method. Wiley. A. Beimel, F. Bergadano, N. Bshouty, E. Kushilevitz & S. Varricchio (1996). On the applications of multiplicity automata in learning. In FOCS '96, 349–358.