On One-Way Functions from NP-Complete Problems

This thesis addresses the computational notion of aesthetics in the framework of multistate two-dimensional cellular automata (2D CA). The measure of complexity is a core concept in computational approaches to aesthetics. Shannon's information theory provided an objective measure of complexity, which led to the emergence of various informational theories of aesthetics. However, entropy fails to take into account the spatial characteristics of 2D patterns; these characteristics are fundamental in addressing the aesthetic problem, in general, and of CA-generated patterns, in particular. This thesis proposes two empirically evaluated alternative measures of complexity, taking into account the spatial characteristics of 2D patterns and experimental studies on human aesthetic perception in the visual domain. The measures are extended to robustly quantify the complexity of multi-state 2D CA-generated patterns. The first model, spatial complexity, is based on the probabilistic spatial distribution of homogeneous/heterogeneous neighbouring cells over the lattice of a multi-state 2D cellular automaton. The second model is based on algorithmic information theory (Kolmogorov complexity) which is extended to estimate the complexity of 2D patterns. The spatial complexity measure presents performance advantage over information-theoretic models, specifically in discriminating symmetries and the orientation in CA-generated patterns, enabling more accurate measurement of complexity in relation to aesthetic evaluations of 2D patterns. A series of experimental stimuli with various structural characteristics and levels of complexity were generated by seeding 3-state 2D CA with different initial configurations for psychological experiments. The results of experimentation demonstrate the presence of correlation between spatial complexity measures and aesthetic judgements of experimental stimuli. The same results were obtained for the estimations of Kolmogorov complexity of experimental stimuli.

Liu and Pass (FOCS’20) recently demonstrated an equivalence between the existence of one-way functions (OWFs) and mild average-case hardness of the time-bounded Kolmogorov complexity problem. In this work, we establish a similar equivalence but to a different form of time-bounded Kolmogorov Complexity—namely, Levin’s notion of Kolmogorov Complexity—whose hardness is closely related to the problem of whether \(\mathsf{EXP}\ne \mathsf{BPP}\). In more detail, let Kt(x) denote the Levin-Kolmogorov Complexity of the string x; that is, \(Kt(x) = \min _{{\varPi }\in \{0,1\}^*, t \in \mathbb {N}}\{|{\varPi }| + \lceil \log t \rceil : U({\varPi }, 1^t) = x\}\), where U is a universal Turing machine, and \(U({\varPi },1^t)\) denotes the output of the program \(\varPi \) after t steps, and let \(\mathsf{MKtP}\) denote the language of pairs (x, k) having the property that \(Kt(x) \le k\). We demonstrate that: \(\mathsf{MKtP}\notin \mathsf{Heur}_{\mathsf{neg}}\mathsf{BPP}\) (i.e., \(\mathsf{MKtP}\) is infinitely-often two-sided error mildly average-case hard) iff infinitely-often OWFs exist. \(\mathsf{MKtP}\notin \mathsf{Avg}_{\mathsf{neg}}\mathsf{BPP}\) (i.e., \(\mathsf{MKtP}\) is infinitely-often errorless mildly average-case hard) iff \(\mathsf{EXP}\ne \mathsf{BPP}\).

We show that the existence of one-way functions is necessary and sufficient for the existence of pseudo-random generators in the following sense. Let ƒ be an easily computable function such that when x is chosen randomly: (1) from ƒ(x) it is hard to recover an x1 with ƒ(x1) = ƒ(x) by a small circuit, or; (2) ƒ has small degeneracy and from ƒ(x) it is hard to recover x by a fast algorithm. From one-way functions of type (1) or (2) we show how to construct pseudo-random generators secure against small circuits or fast algorithms, respectively, and vice-versa. Previous results show how to construct pseudo-random generators from one-way functions that have special properties ([Blum, Micali 82], [Yao 82], [Levin 85], [Goldreich, Krawczyk, Luby 88]).We use the results of [Goldreich, Levin 89] in an essential way.

We give a new construction of pseudorandom generators from any one-way function. The construction achieves better parameters and is simpler than that given in the seminal work of H\aastad, Impagliazzo, Levin, and Luby [SIAM J. Comput., 28 (1999), pp. 1364--1396]. The key to our construction is a new notion of next-block pseudoentropy, which is inspired by the notion of “inaccessible entropy” recently introduced in [I. Haitner, O. Reingold, S. Vadhan, and H. Wee, Proceedings of the $41$st Annual ACM Symposium on Theory of Computing (STOC), 2009, pp. 611--620]. An additional advantage over previous constructions is that our pseudorandom generators are parallelizable and invoke the one-way function in a nonadaptive manner. Using [B. Applebaum, Y. Ishai, and E. Kushilevitz, SIAM J. Comput., 36 (2006), pp. 845--888], this implies the existence of pseudorandom generators in NC$^0$ based on the existence of one-way functions in NC$^1$.

Unravelling the structure of genotype–phenotype (GP) maps is an important problem in biology. Recently, arguments inspired by algorithmic information theory (AIT) and Kolmogorov complexity have been invoked to uncover simplicity bias in GP maps, an exponentially decaying upper bound in phenotype probability with the increasing phenotype descriptional complexity. This means that phenotypes with many genotypes assigned via the GP map must be simple, while complex phenotypes must have few genotypes assigned. Here, we use similar arguments to bound the probability P(x → y) that phenotype x, upon random genetic mutation, transitions to phenotype y. The bound is , where is the estimated conditional complexity of y given x, quantifying how much extra information is required to make y given access to x. This upper bound is related to the conditional form of algorithmic probability from AIT. We demonstrate the practical applicability of our derived bound by predicting phenotype transition probabilities (and other related quantities) in simulations of RNA and protein secondary structures. Our work contributes to a general mathematical understanding of GP maps and may facilitate the prediction of transition probabilities directly from examining phenotype themselves, without utilizing detailed knowledge of the GP map.

One of the central lines of cryptographic research is identifying the weakest assumptions required for the construction of secure primitives. In the context of group signatures the gap between what is known to be necessary (one-way functions) and what is known to be sufficient (trapdoor permutations) is quite large. In this paper, we provide the first step towards closing this gap by showing that the existence of secure group signature schemes implies the existence of secure public-key encryption schemes. Our result shows that the construction of secure group signature schemes based solely on the existence of one-way functions is unlikely. This is in contrast to what is known for standard signature schemes, which can be constructed from any one-way function.

Uncomputability and undecidability in economic theory

We study characterizations of one-way functions in terms of time-bounded Kolmogorov complexity. As the main contribution, we propose definitions for strong and weak Kolmogorov one-way functions and show that these are equivalent to classical strong and weak one-way functions, respectively. The new definitions were motivated by the fact that the expected value approach is not able to characterize strong one-way functions as we prove in the paper.

New NP-complete problem, named as multivariate quadratic power (MQP) problem, is presented. It is based on solution of multivariate quadratic power system of equations over the semigroup Zn, denoted by MQP(Zn), where n is positive integer. Two sequential polynomial-time reductions from known NP-complete multivariate quadratic (MQ) problem over the field Z2, i.e. MQ(Z2) to MQP(Zn) are constructed. It is proved that certain restricted MQP(Zn) problem over some subgroup of Zn is equivalent to MQ(Z2) problem. This allow us to prove that MQP(Zn) is NP-complete also.MQP problem is linked to some author’s previously declared matrix power function (MPF) used for several cryptographic protocols construction. Obtained NP-complete problem will be used to create new candidate one-way function (OWF) based on MPF for new cryptographic primitives’ construction.DOI: http://dx.doi.org/10.5755/j01.itc.41.1.821

Although algorithmic information theory provides a measure of the information content of string of characters, problems of noise and noncomputability emerge. However, if pattern in a noisy string is recognized by reference to a set of similar strings, this article shows that a compressed algorithmic description of a noisy string is possible and illustrates this with some simple examples. The article also shows that algorithmic information theory can quantify the information in complex organized systems where pattern is nested within pattern. © 2006 Wiley Periodicals, Inc. Complexity 12: 52–58, 2006

Unknowability in Mathematics, Biology, and Physics Gregory J. Chaitin (bio) there are three fields in mathematics that deal with unknowability; in chronological order, they are probability theory, metamathematics, and algorithmic information theory. The theory of probabilities began in a series of letters between Pascal and Fermat. One of the first books in this field is A Philosophical Essay on Probabilities by Pierre-Simon Laplace, published in 1814, which is still very much worth reading. The theory of probability is about indeterminate events, but it is not about mathematical facts, which are always black or white, not grey, and which are necessary, not contingent, truths. However, the field of meta mathematics is actually about unknowability within mathematics itself—and a fascinating piece of intellectual history associated with the names David Hilbert, Kurt Gödel, and Alan Turing. These three gentlemen are mentioned elsewhere in this volume, so I will not dwell on them. The newest mathematical field dealing with unknowability is algorithmic information theory. The star of this theory is a real number, the halting probability Omega, which is defined as the probability that a self-contained computer program chosen at random will eventually halt. If all programs halted, the halting probability would be one. If no program halted, the halting probability would be zero. And since some programs eventually halt and others don't, Omega is actually between zero and one. [End Page 107] Omega has a rather straightforward mathematical definition, and it is a single, well-defined real number. Yet its numerical value is maximally unknowable. The base-two bits of the numerical value of the halting probability Omega provide a perfect simulation within pure mathematics of independent tosses of a fair coin. Whether an individual bit is a zero or a one is, to God or to an infinite mind, a single, well-determined individual mathematical truth, but to us down here, it appears maximally unknowable and very much like the result of independent tosses of a fair coin. Hilbert had hoped that there would be a theory of everything for pure mathematics, but Omega shows that pure math is more like biology, the domain of the irreducibly complex, than it is like fundamental physics, where there is still hope of finding a small set of simple equations that determine the universe. So that brings us to biology, a very messy, complicated field. I have a wife and a child, so I love messy biology, but from a mathematical point of view, biology looks rather intractable, rather opaque. Will we ever be able to prove mathematically that life must evolve? Does Darwin's theory provably account for the spectacular biological creativity that led to human beings, to you and me? Can one prove that random mutations and natural selection are enough to account for the richness and diversity of our biosphere? There is a highly developed field called systems biology that deals with computer simulations of complex biological systems, but there are very few mathematical proofs in biology. What can be done about this? Well, in a remarkable 1948 lecture published in 1951 and remarkably forgotten, The General and Logical Theory of Automata, John von Neumann identified the fundamental mathematical idea hidden in biology. That's the idea of software, which accounts both for the plasticity of computer technology and its overwhelming success, and for the plasticity of the biosphere with its spectacular diversity of forms. To put it bluntly, nature invented software millennia before humanity did. Biology deals with extremely ancient software. "Evodevo," or evolutionary developmental biology, is a kind of software archeology. [End Page 108] The new field I call "metabiology" recognizes the impossibility of a fundamental mathematical theory dealing with the effect of random mutations on natural software, on DNA, so instead it proposes studying the effect of random mutations on artificial software, on computer programs. That, in fact, turns out to be tractable mathematically, and it is probably as close as one can get to a theoretical biology as deeply impregnated with mathematics as theoretical physics already is. When I was a visiting professor at the Federal University of Rio de Janeiro, I sketched out these ideas, and they were published in a little...

Algorithmic complexity is a sub-field of Claude Shannon’s Information Theory (1948) and is known by various names as algorithmic information theory, kolmogorov complexity, and algorithmic complexity to name a few such monikers and was developed by R. J. Solomonoff, A. N. Kolmogorov, and G. J. Chaitin in the 1960’s (Chaitin, 1982).

Discussions about information centered, either in favor or against, the use of Shannon’s information theory are not only ill-suited but cannot help the discussion unless it is left behind just as the theory of randomness did early in the 1960s with the theory of algorithmic information. While it is true that physics, and many other areas, have been painfully slow at moving away from Shannon entropy and still can find a wide range of applications for it, we will show how eventually more powerful generating and predictive data-driven models will—and should—replace it. I will argue that discussions in Philosophy of Information should be thus one step ahead instead of several behind, guiding not only the philosophical discussion but also leading and steering scientific attention. I will explain the relevance of algorithmic complexity as a salient property at the core of the scientific method, especially in the challenge of causality discovery, and how fears against moving away from Entropy based upon arguments of uncomputability are unfounded, as it precludes progress and embraces defeat.

This paper presents a philosophical and theoretical follow-up to the author’s recent work on hallucination as an inevitable byproduct of intelligence in LLMs. While the original paper argued that hallucination is not merely a failure mode but an intrinsic property of intelligent behavior in LLMs, this paper extends that thesis by exploring whether this inevitability could ever be overturned.The central question examined is: Could hallucination be eliminated, if lossless abstraction were made possible? We analyze this hypothetical from first principles and argue that any such goal runs counter to the foundational limits imposed by algorithmic information theory, particularly Kolmogorov complexity, Chaitin’s incompleteness theorem, and Shannon’s entropy limit. Together, these frameworks demonstrate that any sufficiently intelligent system must operate through lossy compression entailing inherent information loss, ambiguity, and semantic distortion.This implies that hallucination is not a superficial flaw to be engineered away, but rather a structural consequence of intelligence operating under finite constraints. We conclude by reflecting on the implications of this claim for AI design, evaluation, and ethics, proposing a shift toward hallucination-aware architectures and epistemically humble interfaces.

We give a construction of non-malleable statistically hiding commitments based on the existence of one-way functions. Our construction employs statistically hiding commitment schemes recently proposed by Haitner and Reingold [1], and special-sound WI proofs. Our proof of security relies on the message scheduling technique introduced by Dolev, Dwork and Naor [2], and requires only the use of black-box techniques.