Pub Date : 2025-08-22DOI: 10.1109/TIT.2025.3598529
{"title":"TechRxiv: Share Your Preprint Research with the World!","authors":"","doi":"10.1109/TIT.2025.3598529","DOIUrl":"https://doi.org/10.1109/TIT.2025.3598529","url":null,"abstract":"","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 9","pages":"7417-7417"},"PeriodicalIF":2.9,"publicationDate":"2025-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=11134638","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144891164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-20DOI: 10.1109/TIT.2025.3601047
Lukas Brenner;Christophe Piveteau;David Sutter
Circuit knitting is the process of partitioning large quantum circuits into smaller subcircuits such that the result of the original circuits can be deduced by only running the subcircuits. Such techniques will be crucial for near-term and early fault-tolerant quantum computers, as the limited number of qubits is likely to be a major bottleneck for demonstrating quantum advantage. One typically distinguishes between gate cuts and wire cuts when partitioning a circuit. The cost for any circuit knitting approach scales exponentially in the number of cuts. One possibility to realize a cut is via the quasiprobability simulation technique. In fact, we argue that all existing rigorous circuit knitting techniques can be understood in this framework. Furthermore, we characterize the optimal overhead for wire cuts where the subcircuits can exchange classical information or not. We show that the optimal cost for cutting n wires without and with classical communication between the subcircuits scales as $O(16^{n})$ and $O(4^{n})$ , respectively.
{"title":"Optimal Wire Cutting With Classical Communication","authors":"Lukas Brenner;Christophe Piveteau;David Sutter","doi":"10.1109/TIT.2025.3601047","DOIUrl":"https://doi.org/10.1109/TIT.2025.3601047","url":null,"abstract":"Circuit knitting is the process of partitioning large quantum circuits into smaller subcircuits such that the result of the original circuits can be deduced by only running the subcircuits. Such techniques will be crucial for near-term and early fault-tolerant quantum computers, as the limited number of qubits is likely to be a major bottleneck for demonstrating quantum advantage. One typically distinguishes between gate cuts and wire cuts when partitioning a circuit. The cost for any circuit knitting approach scales exponentially in the number of cuts. One possibility to realize a cut is via the quasiprobability simulation technique. In fact, we argue that all existing rigorous circuit knitting techniques can be understood in this framework. Furthermore, we characterize the optimal overhead for wire cuts where the subcircuits can exchange classical information or not. We show that the optimal cost for cutting <italic>n</i> wires without and with classical communication between the subcircuits scales as <inline-formula> <tex-math>$O(16^{n})$ </tex-math></inline-formula> and <inline-formula> <tex-math>$O(4^{n})$ </tex-math></inline-formula>, respectively.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 10","pages":"7742-7752"},"PeriodicalIF":2.9,"publicationDate":"2025-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-19DOI: 10.1109/TIT.2025.3599674
Salman Habib;Rémi A. Chou;Taejoon Kim
The reconstruction of sparse signals from a limited set of measurements poses a significant challenge as it necessitates a solution to an underdetermined system of linear equations. Compressed sensing (CS) deals with sparse signal reconstruction using techniques such as linear programming (LP) and iterative message passing schemes. The interval passing algorithm (IPA) is an attractive CS approach due to its low complexity when compared to LP. In this paper, we propose a sequential IPA that is inspired by sequential belief propagation decoding of low-density-parity-check (LDPC) codes used for forward error correction in channel coding. In the sequential setting, each check node (CN) in the Tanner graph of an LDPC measurement matrix is scheduled one at a time in every iteration, as opposed to the standard “flooding” interval passing approach in which all CNs are scheduled at once per iteration. The sequential scheme offers a significantly lower message passing complexity compared to flooding IPA on average, and for some measurement matrix and signal sparsity, a complexity reduction of approximately 36% is achieved. We show both analytically and numerically that the reconstruction accuracy of the IPA is not compromised by adopting our sequential scheduling approach.
{"title":"Reduced Complexity Interval Passing for Sparse Signal Recovery","authors":"Salman Habib;Rémi A. Chou;Taejoon Kim","doi":"10.1109/TIT.2025.3599674","DOIUrl":"https://doi.org/10.1109/TIT.2025.3599674","url":null,"abstract":"The reconstruction of sparse signals from a limited set of measurements poses a significant challenge as it necessitates a solution to an underdetermined system of linear equations. Compressed sensing (CS) deals with sparse signal reconstruction using techniques such as linear programming (LP) and iterative message passing schemes. The interval passing algorithm (IPA) is an attractive CS approach due to its low complexity when compared to LP. In this paper, we propose a sequential IPA that is inspired by sequential belief propagation decoding of low-density-parity-check (LDPC) codes used for forward error correction in channel coding. In the sequential setting, each check node (CN) in the Tanner graph of an LDPC measurement matrix is scheduled one at a time in every iteration, as opposed to the standard “flooding” interval passing approach in which all CNs are scheduled at once per iteration. The sequential scheme offers a significantly lower message passing complexity compared to flooding IPA on average, and for some measurement matrix and signal sparsity, a complexity reduction of approximately 36% is achieved. We show both analytically and numerically that the reconstruction accuracy of the IPA is not compromised by adopting our sequential scheduling approach.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 10","pages":"8080-8098"},"PeriodicalIF":2.9,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-18DOI: 10.1109/TIT.2025.3600002
Wengang Jin;Kangquan Li;Longjiang Qu
Linear codes have attracted considerable attention in coding theory and cryptography due to their significant applications in secret sharing schemes, secure two-party computation, and Galois geometries, among others. As two special subclasses of linear codes, minimal linear codes and self-orthogonal linear codes are of particular interest. Constructing linear codes that possess both minimality and self-orthogonality is very interesting. The main purpose of this paper is to construct self-orthogonal minimal linear codes that violate the Ashikhmin-Barg (AB for short) condition over the finite field $mathbb {F}_{p}$ . First, we present several classes of self-orthogonal minimal linear codes violating the AB condition over the finite field $mathbb {F}_{2}$ and determine their weight distributions. Next, for any odd prime p, we construct two classes of self-orthogonal linear codes from p-ary functions, which contain some optimal or almost optimal codes. Finally, based on plateaued functions, we construct two classes of self-orthogonal linear codes that violate the AB condition. Their weight distributions are also provided. To the best of our knowledge, this paper is the first to investigate the constructions of linear codes that violate the AB condition and satisfy self-orthogonality.
{"title":"Several New Classes of Self-Orthogonal Minimal Linear Codes Violating the Ashikhmin–Barg Condition","authors":"Wengang Jin;Kangquan Li;Longjiang Qu","doi":"10.1109/TIT.2025.3600002","DOIUrl":"https://doi.org/10.1109/TIT.2025.3600002","url":null,"abstract":"Linear codes have attracted considerable attention in coding theory and cryptography due to their significant applications in secret sharing schemes, secure two-party computation, and Galois geometries, among others. As two special subclasses of linear codes, minimal linear codes and self-orthogonal linear codes are of particular interest. Constructing linear codes that possess both minimality and self-orthogonality is very interesting. The main purpose of this paper is to construct self-orthogonal minimal linear codes that violate the Ashikhmin-Barg (AB for short) condition over the finite field <inline-formula> <tex-math>$mathbb {F}_{p}$ </tex-math></inline-formula>. First, we present several classes of self-orthogonal minimal linear codes violating the AB condition over the finite field <inline-formula> <tex-math>$mathbb {F}_{2}$ </tex-math></inline-formula> and determine their weight distributions. Next, for any odd prime <italic>p</i>, we construct two classes of self-orthogonal linear codes from <italic>p</i>-ary functions, which contain some optimal or almost optimal codes. Finally, based on plateaued functions, we construct two classes of self-orthogonal linear codes that violate the AB condition. Their weight distributions are also provided. To the best of our knowledge, this paper is the first to investigate the constructions of linear codes that violate the AB condition and satisfy self-orthogonality.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 10","pages":"7699-7714"},"PeriodicalIF":2.9,"publicationDate":"2025-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-13DOI: 10.1109/TIT.2025.3598702
Sjoerd Dirksen;Weilin Li;Johannes Maly
We study direction-of-arrival (DOA) estimation from coarsely quantized data. We focus on a two-step approach which first estimates the signal subspace via covariance estimation and then extracts DOA angles by the ESPRIT algorithm. In particular, we analyze two stochastic quantization schemes which use dithering: a one-bit quantizer combined with rectangular dither and a multi-bit quantizer with triangular dither. For each quantizer, we derive rigorous high probability bounds for the distances between the true and estimated signal subspaces and DOA angles. Using our analysis, we identify scenarios in which subspace and DOA estimation via triangular dithering qualitatively outperforms rectangular dithering. We verify in numerical simulations that our estimates are optimal in their dependence on the smallest non-zero eigenvalue of the target matrix. The resulting subspace estimation guarantees are equally applicable in the analysis of other spectral estimation algorithms and related problems.
{"title":"Subspace and DOA Estimation Under Coarse Quantization","authors":"Sjoerd Dirksen;Weilin Li;Johannes Maly","doi":"10.1109/TIT.2025.3598702","DOIUrl":"https://doi.org/10.1109/TIT.2025.3598702","url":null,"abstract":"We study direction-of-arrival (DOA) estimation from coarsely quantized data. We focus on a two-step approach which first estimates the signal subspace via covariance estimation and then extracts DOA angles by the ESPRIT algorithm. In particular, we analyze two stochastic quantization schemes which use dithering: a one-bit quantizer combined with rectangular dither and a multi-bit quantizer with triangular dither. For each quantizer, we derive rigorous high probability bounds for the distances between the true and estimated signal subspaces and DOA angles. Using our analysis, we identify scenarios in which subspace and DOA estimation via triangular dithering qualitatively outperforms rectangular dithering. We verify in numerical simulations that our estimates are optimal in their dependence on the smallest non-zero eigenvalue of the target matrix. The resulting subspace estimation guarantees are equally applicable in the analysis of other spectral estimation algorithms and related problems.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 10","pages":"8149-8168"},"PeriodicalIF":2.9,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-13DOI: 10.1109/TIT.2025.3598296
Han Qi;Haochen Yang;Qiaosheng Zhang;Zhuoran Yang
We study the problem of reinforcement learning from human feedback (RLHF), a critical problem in training large language models, from a theoretical perspective. Our main contribution is the design of novel sample-efficient RLHF algorithms based on information-directed sampling (IDS), an online decision-making principle inspired by information theory. Our algorithms maximize the sum of the value function and a mutual information term that encourages exploration of the unknown environment (which quantifies the information gained about the environment through observed human feedback data). To tackle the challenge of large state spaces and improve sample efficiency, we construct a simplified surrogate environment and introduce a novel distance measure (named the $ell _{g}$ -distance), enabling our IDS-based algorithm to achieve a Bayesian regret upper bound of order $O(H^{3/2}sqrt {log (K(epsilon)) T})$ , where H is the episode length, T is the number of episode and $K(epsilon)$ is related to the covering number of the environment. Specializing to the tabular settings, this regret bound is of order $tilde {O}(H^{2}sqrt {SAT})$ , where S and A are the numbers of states and actions. Finally, we propose an Approximate-IDS algorithm that is computationally more efficient while maintaining nearly the same sample efficiency. The design principle of this approximate algorithm is not only effective in RLHF settings but also applicable to the standard RL framework. Moreover, our work showcases the value of information theory in reinforcement learning and in the training of large language models.
{"title":"Sample-Efficient Reinforcement Learning From Human Feedback via Information-Directed Sampling","authors":"Han Qi;Haochen Yang;Qiaosheng Zhang;Zhuoran Yang","doi":"10.1109/TIT.2025.3598296","DOIUrl":"https://doi.org/10.1109/TIT.2025.3598296","url":null,"abstract":"We study the problem of reinforcement learning from human feedback (RLHF), a critical problem in training large language models, from a theoretical perspective. Our main contribution is the design of novel sample-efficient RLHF algorithms based on information-directed sampling (IDS), an online decision-making principle inspired by information theory. Our algorithms maximize the sum of the value function and a mutual information term that encourages exploration of the unknown environment (which quantifies the information gained about the environment through observed human feedback data). To tackle the challenge of large state spaces and improve sample efficiency, we construct a simplified <italic>surrogate environment</i> and introduce a novel distance measure (named the <inline-formula> <tex-math>$ell _{g}$ </tex-math></inline-formula><italic>-distance</i>), enabling our IDS-based algorithm to achieve a Bayesian regret upper bound of order <inline-formula> <tex-math>$O(H^{3/2}sqrt {log (K(epsilon)) T})$ </tex-math></inline-formula>, where <italic>H</i> is the episode length, <italic>T</i> is the number of episode and <inline-formula> <tex-math>$K(epsilon)$ </tex-math></inline-formula> is related to the covering number of the environment. Specializing to the tabular settings, this regret bound is of order <inline-formula> <tex-math>$tilde {O}(H^{2}sqrt {SAT})$ </tex-math></inline-formula>, where <italic>S</i> and <italic>A</i> are the numbers of states and actions. Finally, we propose an Approximate-IDS algorithm that is computationally more efficient while maintaining nearly the same sample efficiency. The design principle of this approximate algorithm is not only effective in RLHF settings but also applicable to the standard RL framework. Moreover, our work showcases the value of information theory in reinforcement learning and in the training of large language models.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 10","pages":"7942-7958"},"PeriodicalIF":2.9,"publicationDate":"2025-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-12DOI: 10.1109/TIT.2025.3597943
Masahito Hayashi
We generalize the generalized Arimoto-Blahut algorithm to a general function defined over Bregman-divergence system. In existing methods, when linear constraints are imposed, each iteration needs to solve a convex minimization. Exploiting our obtained algorithm, we propose a minimization-free-iteration algorithm. This algorithm can be applied to classical and quantum rate-distortion theory. We numerically apply our method to the derivation of the optimal conditional distribution in the rate-distortion theory.
{"title":"Bregman-Divergence-Based Arimoto-Blahut Algorithm","authors":"Masahito Hayashi","doi":"10.1109/TIT.2025.3597943","DOIUrl":"https://doi.org/10.1109/TIT.2025.3597943","url":null,"abstract":"We generalize the generalized Arimoto-Blahut algorithm to a general function defined over Bregman-divergence system. In existing methods, when linear constraints are imposed, each iteration needs to solve a convex minimization. Exploiting our obtained algorithm, we propose a minimization-free-iteration algorithm. This algorithm can be applied to classical and quantum rate-distortion theory. We numerically apply our method to the derivation of the optimal conditional distribution in the rate-distortion theory.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 10","pages":"7788-7801"},"PeriodicalIF":2.9,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-11DOI: 10.1109/TIT.2025.3597546
Hojat Allah Salehi;Farhad Shirani
This work presents a Fourier analysis framework for the non-interactive source simulation (NISS) problem. Two distributed agents observe a pair of sequences $X^{d}$ and $Y^{d}$ drawn according to a joint distribution $P_{X^{d}Y^{d}}$ . The agents aim to generate outputs $U=f_{d}(X^{d})$ and $V=g_{d}(Y^{d})$ with a joint distribution sufficiently close in total variation to a target distribution $Q_{UV}$ . Existing works have shown that the NISS problem with finite-alphabet outputs is decidable. For the binary-output NISS, an upper-bound to the input complexity was derived which is $Oleft ({{exp mathrm {poly}left ({{frac {1}{epsilon }}}right)}}right)$ . In this work, the input complexity and algorithm design are addressed in several classes of NISS scenarios. For binary-output NISS scenarios with doubly-symmetric binary inputs, it is shown that the input complexity is $Theta left ({{log {frac {1}{epsilon }}}}right)$ , thus providing a super-exponential improvement in input complexity. An explicit characterization of the simulating pair of functions is provided. For general finite-input scenarios, a constructive algorithm is introduced that explicitly finds the simulating functions $(f_{d}(X^{d}),g_{d}(Y^{d}))$ . The approach relies on a novel Fourier analysis framework. Various numerical simulations of NISS scenarios with IID inputs are provided. Furthermore, to illustrate the general applicability of the Fourier framework, several examples with non-IID inputs, including entanglement-assisted NISS and NISS with Markovian inputs are provided.
{"title":"On Non-Interactive Simulation of Distributed Sources With Finite Alphabets","authors":"Hojat Allah Salehi;Farhad Shirani","doi":"10.1109/TIT.2025.3597546","DOIUrl":"https://doi.org/10.1109/TIT.2025.3597546","url":null,"abstract":"This work presents a Fourier analysis framework for the non-interactive source simulation (NISS) problem. Two distributed agents observe a pair of sequences <inline-formula> <tex-math>$X^{d}$ </tex-math></inline-formula> and <inline-formula> <tex-math>$Y^{d}$ </tex-math></inline-formula> drawn according to a joint distribution <inline-formula> <tex-math>$P_{X^{d}Y^{d}}$ </tex-math></inline-formula>. The agents aim to generate outputs <inline-formula> <tex-math>$U=f_{d}(X^{d})$ </tex-math></inline-formula> and <inline-formula> <tex-math>$V=g_{d}(Y^{d})$ </tex-math></inline-formula> with a joint distribution sufficiently close in total variation to a target distribution <inline-formula> <tex-math>$Q_{UV}$ </tex-math></inline-formula>. Existing works have shown that the NISS problem with finite-alphabet outputs is decidable. For the binary-output NISS, an upper-bound to the input complexity was derived which is <inline-formula> <tex-math>$Oleft ({{exp mathrm {poly}left ({{frac {1}{epsilon }}}right)}}right)$ </tex-math></inline-formula>. In this work, the input complexity and algorithm design are addressed in several classes of NISS scenarios. For binary-output NISS scenarios with doubly-symmetric binary inputs, it is shown that the input complexity is <inline-formula> <tex-math>$Theta left ({{log {frac {1}{epsilon }}}}right)$ </tex-math></inline-formula>, thus providing a super-exponential improvement in input complexity. An explicit characterization of the simulating pair of functions is provided. For general finite-input scenarios, a constructive algorithm is introduced that explicitly finds the simulating functions <inline-formula> <tex-math>$(f_{d}(X^{d}),g_{d}(Y^{d}))$ </tex-math></inline-formula>. The approach relies on a novel Fourier analysis framework. Various numerical simulations of NISS scenarios with IID inputs are provided. Furthermore, to illustrate the general applicability of the Fourier framework, several examples with non-IID inputs, including entanglement-assisted NISS and NISS with Markovian inputs are provided.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 10","pages":"8048-8079"},"PeriodicalIF":2.9,"publicationDate":"2025-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-08DOI: 10.1109/TIT.2025.3597092
Marcus Hutter
The domain-independent universal Normalized Information Distance based on Kolmogorov complexity has been (in approximate form) successfully applied to a variety of difficult clustering problems. In this paper we investigate theoretical properties of the un-normalized algorithmic information distance $d_{K}$ . The main question we are asking in this work is what properties this curious distance has, besides being a metric. We show that many (in)finite-dimensional spaces can(not) be isometrically scale-embedded into the space of finite strings with metric $d_{K}$ . We also show that $d_{K}$ is not an Euclidean distance, but any finite set of points in Euclidean space can be scale-embedded into $({0,1}^{*},d_{K})$ . A major contribution is the development of the necessary framework and tools for finding more (interesting) properties of $d_{K}$ in future, and to state several open problems.
{"title":"Properties of Algorithmic Information Distance","authors":"Marcus Hutter","doi":"10.1109/TIT.2025.3597092","DOIUrl":"https://doi.org/10.1109/TIT.2025.3597092","url":null,"abstract":"The domain-independent universal Normalized Information Distance based on Kolmogorov complexity has been (in approximate form) successfully applied to a variety of difficult clustering problems. In this paper we investigate theoretical properties of the un-normalized algorithmic information distance <inline-formula> <tex-math>$d_{K}$ </tex-math></inline-formula>. The main question we are asking in this work is what properties this curious distance has, besides being a metric. We show that many (in)finite-dimensional spaces can(not) be isometrically scale-embedded into the space of finite strings with metric <inline-formula> <tex-math>$d_{K}$ </tex-math></inline-formula>. We also show that <inline-formula> <tex-math>$d_{K}$ </tex-math></inline-formula> is not an Euclidean distance, but any finite set of points in Euclidean space can be scale-embedded into <inline-formula> <tex-math>$({0,1}^{*},d_{K})$ </tex-math></inline-formula>. A major contribution is the development of the necessary framework and tools for finding more (interesting) properties of <inline-formula> <tex-math>$d_{K}$ </tex-math></inline-formula> in future, and to state several open problems.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 10","pages":"7540-7554"},"PeriodicalIF":2.9,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110241","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-08DOI: 10.1109/TIT.2025.3597162
Nurdagül Anbar;Tekgül Kalaycı;Alev Topuzoğlu
We introduce a new concept, the APN-defect, which can be thought of as measuring the distance of a given function $G:mathbb {F}_{2^{n}} rightarrow mathbb {F}_{2^{n}}$ to the set of almost perfect nonlinear (APN) functions. This concept is motivated by the detailed analysis of the differential behaviour of non-APN functions (of low differential uniformity) G using the so-called difference squares. Indeed, the insight into some structural qualities of S-boxes provided by this new approach is particularly useful in the light of recent refinements of differential cryptanalysis. We describe the relations between the APN-defect and other current concepts of similar nature. Values of APN-defect for several classes of functions of interest, including Dembowski-Ostrom polynomials are given. This enables one to identify the quasi-APN ones, i.e., those with favourable differential behavior. The difference square corresponding to a modification of the inverse function is determined, its APN-defect depending on n is evaluated, the partial quadruple system associated to it is described, and the implications are discussed. In the forthcoming second part of this work we further examine the APN-defect of modifications of the inverse function and address some questions concerning CCZ-equivalence. We also study modifications of classes of functions of low differential uniformity over infinitely many extensions of $mathbb {F}_{2^{n}}$ and present quantitative results on their differential behaviour.
{"title":"Analysis of Functions of Low Differential Uniformity in Characteristic 2: A New Approach (I)","authors":"Nurdagül Anbar;Tekgül Kalaycı;Alev Topuzoğlu","doi":"10.1109/TIT.2025.3597162","DOIUrl":"https://doi.org/10.1109/TIT.2025.3597162","url":null,"abstract":"We introduce a new concept, the <italic>APN-defect</i>, which can be thought of as measuring the distance of a given function <inline-formula> <tex-math>$G:mathbb {F}_{2^{n}} rightarrow mathbb {F}_{2^{n}}$ </tex-math></inline-formula> to the set of almost perfect nonlinear (APN) functions. This concept is motivated by the detailed analysis of the differential behaviour of non-APN functions (of low differential uniformity) <italic>G</i> using the so-called <italic>difference squares</i>. Indeed, the insight into some structural qualities of S-boxes provided by this new approach is particularly useful in the light of recent refinements of differential cryptanalysis. We describe the relations between the APN-defect and other current concepts of similar nature. Values of APN-defect for several classes of functions of interest, including Dembowski-Ostrom polynomials are given. This enables one to identify the <italic>quasi-APN</i> ones, i.e., those with favourable differential behavior. The difference square corresponding to a modification of the inverse function is determined, its APN-defect depending on <italic>n</i> is evaluated, the partial quadruple system associated to it is described, and the implications are discussed. In the forthcoming second part of this work we further examine the APN-defect of modifications of the inverse function and address some questions concerning CCZ-equivalence. We also study modifications of classes of functions of low differential uniformity over infinitely many extensions of <inline-formula> <tex-math>$mathbb {F}_{2^{n}}$ </tex-math></inline-formula> and present quantitative results on their differential behaviour.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"71 10","pages":"8002-8016"},"PeriodicalIF":2.9,"publicationDate":"2025-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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