Pub Date : 2024-09-17DOI: 10.1109/TIT.2024.3462712
Xue Jia;Fengwei Li;Huan Sun;Qin Yue
Classical Goppa codes are a well-known class of codes with applications in code-based cryptography, which are a special case of alternant codes. Many papers are devoted to the search for Goppa codes with a cyclic extension or with a cyclic parity-check subcode. Let $Bbb F_{q}$ be a finite field with $q=2^{l}$ elements, where l is a positive integer. In this paper, we determine all the generator polynomials of cyclic expurgated or extended Goppa codes under some prescribed permutations induced by the projective general linear automorphism $A in PGL_{2}(Bbb F_{q})$ . Moreover, we provide some examples to support our findings.
{"title":"Generator Polynomials of Cyclic Expurgated or Extended Goppa Codes","authors":"Xue Jia;Fengwei Li;Huan Sun;Qin Yue","doi":"10.1109/TIT.2024.3462712","DOIUrl":"10.1109/TIT.2024.3462712","url":null,"abstract":"Classical Goppa codes are a well-known class of codes with applications in code-based cryptography, which are a special case of alternant codes. Many papers are devoted to the search for Goppa codes with a cyclic extension or with a cyclic parity-check subcode. Let \u0000<inline-formula> <tex-math>$Bbb F_{q}$ </tex-math></inline-formula>\u0000 be a finite field with \u0000<inline-formula> <tex-math>$q=2^{l}$ </tex-math></inline-formula>\u0000 elements, where l is a positive integer. In this paper, we determine all the generator polynomials of cyclic expurgated or extended Goppa codes under some prescribed permutations induced by the projective general linear automorphism \u0000<inline-formula> <tex-math>$A in PGL_{2}(Bbb F_{q})$ </tex-math></inline-formula>\u0000. Moreover, we provide some examples to support our findings.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"7711-7722"},"PeriodicalIF":2.2,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252676","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 : 2024-09-17DOI: 10.1109/TIT.2024.3455153
{"title":"IEEE Transactions on Information Theory Publication Information","authors":"","doi":"10.1109/TIT.2024.3455153","DOIUrl":"10.1109/TIT.2024.3455153","url":null,"abstract":"","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 10","pages":"C2-C2"},"PeriodicalIF":2.2,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10682504","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252682","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 : 2024-09-13DOI: 10.1109/TIT.2024.3460474
Henri Hentilä;Yanina Y. Shkel;Visa Koivunen
We study communication-constrained secret key generation, where two legitimate parties would like to generate a secret key using communication subject to a rate constraint. The problem is studied in the finite-blocklength regime. In this regime, the use of auxiliary random variables subject to Markov chain conditions in the corresponding asymptotic bounds has proven to make most existing proof techniques insufficient. However, two recently proposed proof techniques – one for the achievability side based on Poisson matching, and another for the converse side based on reverse hypercontractivity – allow us to overcome these issues to some extent. Based on these techniques, novel one-shot and second-order achievability and converse bounds are derived for the problem. While the second-order bounds do not coincide, leaving a precise second-order characterization of the problem an open issue, they improve upon the previously known tightest bounds. The second-order bounds are demonstrated for two simple sources: the binary symmetric source and the Gaussian symmetric source. For the binary source, we find that the gap between the two bounds is mainly due to an unwanted constant in the converse bound, and the non-convexity of the achievability bound.
{"title":"Communication-Constrained Secret Key Generation: Second-Order Bounds","authors":"Henri Hentilä;Yanina Y. Shkel;Visa Koivunen","doi":"10.1109/TIT.2024.3460474","DOIUrl":"10.1109/TIT.2024.3460474","url":null,"abstract":"We study communication-constrained secret key generation, where two legitimate parties would like to generate a secret key using communication subject to a rate constraint. The problem is studied in the finite-blocklength regime. In this regime, the use of auxiliary random variables subject to Markov chain conditions in the corresponding asymptotic bounds has proven to make most existing proof techniques insufficient. However, two recently proposed proof techniques – one for the achievability side based on Poisson matching, and another for the converse side based on reverse hypercontractivity – allow us to overcome these issues to some extent. Based on these techniques, novel one-shot and second-order achievability and converse bounds are derived for the problem. While the second-order bounds do not coincide, leaving a precise second-order characterization of the problem an open issue, they improve upon the previously known tightest bounds. The second-order bounds are demonstrated for two simple sources: the binary symmetric source and the Gaussian symmetric source. For the binary source, we find that the gap between the two bounds is mainly due to an unwanted constant in the converse bound, and the non-convexity of the achievability bound.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"8180-8203"},"PeriodicalIF":2.2,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142252677","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 : 2024-09-13DOI: 10.1109/TIT.2024.3460189
Huaning Liu;Arne Winterhof
In the most interesting case of safe prime powers q, Gómez and Winterhof showed that a subfamily of the family of Golomb Costas permutations of ${1,2,ldots,q-2}$ of size $varphi (q-1)$ has maximal cross-correlation of order of magnitude at most $q^{1/2}$ . In this paper we study a larger family of Golomb Costas permutations and prove a weaker bound on its maximal cross-correlation. Considering the whole family of Golomb Costas permutations we show that large cross-correlations are very rare. Finally, we collect several conditions for a small cross-correlation of two Costas permutations. Our main tools are the Weil bound and the Szemerédi-Trotter theorem for finite fields.
{"title":"On the Cross-Correlation of Golomb Costas Permutations","authors":"Huaning Liu;Arne Winterhof","doi":"10.1109/TIT.2024.3460189","DOIUrl":"10.1109/TIT.2024.3460189","url":null,"abstract":"In the most interesting case of safe prime powers q, Gómez and Winterhof showed that a subfamily of the family of Golomb Costas permutations of \u0000<inline-formula> <tex-math>${1,2,ldots,q-2}$ </tex-math></inline-formula>\u0000 of size \u0000<inline-formula> <tex-math>$varphi (q-1)$ </tex-math></inline-formula>\u0000 has maximal cross-correlation of order of magnitude at most \u0000<inline-formula> <tex-math>$q^{1/2}$ </tex-math></inline-formula>\u0000. In this paper we study a larger family of Golomb Costas permutations and prove a weaker bound on its maximal cross-correlation. Considering the whole family of Golomb Costas permutations we show that large cross-correlations are very rare. Finally, we collect several conditions for a small cross-correlation of two Costas permutations. Our main tools are the Weil bound and the Szemerédi-Trotter theorem for finite fields.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"7848-7852"},"PeriodicalIF":2.2,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268575","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 : 2024-09-12DOI: 10.1109/TIT.2024.3458953
Rishabh Dudeja;Subhabrata Sen;Yue M. Lu
It has been observed that the performances of many high-dimensional estimation problems are universal with respect to underlying sensing (or design) matrices. Specifically, matrices with markedly different constructions seem to achieve identical performance if they share the same spectral distribution and have “generic” singular vectors. We prove this universality phenomenon for the case of convex regularized least squares (RLS) estimators under a linear regression model with additive Gaussian noise. Our main contributions are two-fold: (1) We introduce a notion of universality classes for sensing matrices, defined through a set of deterministic conditions that fix the spectrum of the sensing matrix and precisely capture the notion of generic singular vectors; (2) We show that for all sensing matrices that lie in the same universality class, the dynamics of the proximal gradient descent algorithm for solving the regression problem, as well as the performance of RLS estimators themselves (under additional strong convexity conditions) are asymptotically identical. In addition to including i.i.d. Gaussian and rotational invariant matrices as special cases, our universality class also contains highly structured, strongly dependent, and even (nearly) deterministic matrices. Examples of the latter include randomly signed versions of incoherent tight frames and randomly subsampled Hadamard transforms. As a consequence of this universality principle, the asymptotic performance of regularized linear regression on many structured matrices constructed with limited randomness can be characterized by using the rotationally invariant ensemble as an equivalent yet mathematically more tractable surrogate.
{"title":"Spectral Universality in Regularized Linear Regression With Nearly Deterministic Sensing Matrices","authors":"Rishabh Dudeja;Subhabrata Sen;Yue M. Lu","doi":"10.1109/TIT.2024.3458953","DOIUrl":"10.1109/TIT.2024.3458953","url":null,"abstract":"It has been observed that the performances of many high-dimensional estimation problems are universal with respect to underlying sensing (or design) matrices. Specifically, matrices with markedly different constructions seem to achieve identical performance if they share the same spectral distribution and have “generic” singular vectors. We prove this universality phenomenon for the case of convex regularized least squares (RLS) estimators under a linear regression model with additive Gaussian noise. Our main contributions are two-fold: (1) We introduce a notion of universality classes for sensing matrices, defined through a set of deterministic conditions that fix the spectrum of the sensing matrix and precisely capture the notion of generic singular vectors; (2) We show that for all sensing matrices that lie in the same universality class, the dynamics of the proximal gradient descent algorithm for solving the regression problem, as well as the performance of RLS estimators themselves (under additional strong convexity conditions) are asymptotically identical. In addition to including i.i.d. Gaussian and rotational invariant matrices as special cases, our universality class also contains highly structured, strongly dependent, and even (nearly) deterministic matrices. Examples of the latter include randomly signed versions of incoherent tight frames and randomly subsampled Hadamard transforms. As a consequence of this universality principle, the asymptotic performance of regularized linear regression on many structured matrices constructed with limited randomness can be characterized by using the rotationally invariant ensemble as an equivalent yet mathematically more tractable surrogate.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"7923-7951"},"PeriodicalIF":2.2,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194903","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 : 2024-09-11DOI: 10.1109/tit.2024.3457802
Chenya Zhao, Yanxun Chang, Tao Feng
{"title":"The existence of optimal (v, 4, 1) optical orthogonal codes achieving the Johnson bound","authors":"Chenya Zhao, Yanxun Chang, Tao Feng","doi":"10.1109/tit.2024.3457802","DOIUrl":"https://doi.org/10.1109/tit.2024.3457802","url":null,"abstract":"","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"10 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194908","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 : 2024-09-10DOI: 10.1109/TIT.2024.3457150
Qian Guo;Thomas Johansson;Vu Nguyen
The problem of decoding random codes is a fundamental problem for code-based cryptography, including recent code-based candidates in the NIST post-quantum standardization process. In this paper, we present a novel Sieving-style Information-set Decoding algorithm, addressing the task of solving the syndrome decoding problem. Our approach involves maintaining a list of weight- $2p$ solution vectors to a partial syndrome decoding problem and then creating new vectors by identifying pairs of vectors that collide in p positions. By gradually increasing the parity-check condition by one and repeating this process iteratively, we find the final solution(s). We show that our novel algorithm performs better than other ISDs in the memory-restricted scenario when applied to McEliece. Notably, in the case of problem instances with very low relative weight, the sieving approach uses significantly less memory compared to other ISD algorithms while being competitive in terms of performance.
{"title":"A New Sieving-Style Information-Set Decoding Algorithm","authors":"Qian Guo;Thomas Johansson;Vu Nguyen","doi":"10.1109/TIT.2024.3457150","DOIUrl":"10.1109/TIT.2024.3457150","url":null,"abstract":"The problem of decoding random codes is a fundamental problem for code-based cryptography, including recent code-based candidates in the NIST post-quantum standardization process. In this paper, we present a novel Sieving-style Information-set Decoding algorithm, addressing the task of solving the syndrome decoding problem. Our approach involves maintaining a list of weight-\u0000<inline-formula> <tex-math>$2p$ </tex-math></inline-formula>\u0000 solution vectors to a partial syndrome decoding problem and then creating new vectors by identifying pairs of vectors that collide in p positions. By gradually increasing the parity-check condition by one and repeating this process iteratively, we find the final solution(s). We show that our novel algorithm performs better than other ISDs in the memory-restricted scenario when applied to McEliece. Notably, in the case of problem instances with very low relative weight, the sieving approach uses significantly less memory compared to other ISD algorithms while being competitive in terms of performance.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"8303-8319"},"PeriodicalIF":2.2,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194905","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 : 2024-09-10DOI: 10.1109/TIT.2024.3455557
Qifa Yan;Zhengchun Zhou;Xiaohu Tang
In this work, a distributed server system composed of multiple servers that holds some coded files and multiple users that are interested in retrieving the linear functions of the files is investigated, where the servers are robust, blind and adversarial in the sense that any J servers can together recover all files, while any I colluding servers cannot obtain any information about the files, and at most A servers maliciously provides erroneous information. In addition, the file library must be secure from a wiretapper who obtains all the signals, and the demands of any subset of users must kept private from the other users and servers, even if they collude. A coding scheme is proposed by incorporating the ideas of Shamir’s secret sharing and key superposition into the framework of Placement Delivery Array (PDA), originally proposed to characterize the single-server coded caching system without any security or privacy constraints. It is shown that PDAs associated to Maddah-Ali and Niesen’s coded caching scheme results in an achievable memory-storage-communication region, such that the storage size and communication load were optimal to within a multiplicative gap, except for the small memory regime when the number of files was smaller than the number of users.
在这项工作中，研究了一个由多个服务器组成的分布式服务器系统，这些服务器保存着一些编码文件，多个用户对检索文件的线性函数感兴趣，服务器具有鲁棒性、盲目性和对抗性，即任意 J 个服务器可以共同恢复所有文件，而任意 I 个串通的服务器无法获得文件的任何信息，最多 A 个服务器恶意提供错误信息。此外，文件库必须确保不被窃听者获取所有信号，任何用户子集的需求都必须对其他用户和服务器保密，即使他们串通一气。通过将沙米尔秘密共享和密钥叠加的思想融入放置传送阵列（PDA）框架，提出了一种编码方案，该方案最初是为了描述没有任何安全或隐私限制的单服务器编码缓存系统。研究表明，与 Maddah-Ali 和 Niesen 的编码缓存方案相关联的 PDA 会产生一个可实现的内存-存储-通信区域，从而使存储大小和通信负载在乘法差距内达到最佳状态，但文件数量少于用户数量的小内存系统除外。
{"title":"Robust, Secure, and Private Cache-Aided Scalar Linear Function Retrieval From Distributed System With Blind and Adversarial Servers","authors":"Qifa Yan;Zhengchun Zhou;Xiaohu Tang","doi":"10.1109/TIT.2024.3455557","DOIUrl":"10.1109/TIT.2024.3455557","url":null,"abstract":"In this work, a distributed server system composed of multiple servers that holds some coded files and multiple users that are interested in retrieving the linear functions of the files is investigated, where the servers are robust, blind and adversarial in the sense that any J servers can together recover all files, while any I colluding servers cannot obtain any information about the files, and at most A servers maliciously provides erroneous information. In addition, the file library must be secure from a wiretapper who obtains all the signals, and the demands of any subset of users must kept private from the other users and servers, even if they collude. A coding scheme is proposed by incorporating the ideas of Shamir’s secret sharing and key superposition into the framework of Placement Delivery Array (PDA), originally proposed to characterize the single-server coded caching system without any security or privacy constraints. It is shown that PDAs associated to Maddah-Ali and Niesen’s coded caching scheme results in an achievable memory-storage-communication region, such that the storage size and communication load were optimal to within a multiplicative gap, except for the small memory regime when the number of files was smaller than the number of users.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"8237-8250"},"PeriodicalIF":2.2,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194917","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 : 2024-09-10DOI: 10.1109/TIT.2024.3455228
Pablo Pascual Cobo;Kuan Hsieh;Ramji Venkataramanan
We consider the problem of signal estimation in a generalized linear model (GLM). GLMs include many canonical problems in statistical estimation, such as linear regression, phase retrieval, and 1-bit compressed sensing. Recent work has precisely characterized the asymptotic minimum mean-squared error (MMSE) for GLMs with i.i.d. Gaussian sensing matrices. However, in many models there is a significant gap between the MMSE and the performance of the best known feasible estimators. We address this issue by considering GLMs defined via spatially coupled sensing matrices. We propose an efficient approximate message passing (AMP) algorithm for estimation and prove that with a simple choice of spatially coupled design, the MSE of a carefully tuned AMP estimator approaches the asymptotic MMSE as the dimensions of the signal and the observation grow proportionally. To prove the result, we first rigorously characterize the asymptotic performance of AMP for a GLM with a generic spatially coupled design. This characterization is in terms of a deterministic recursion (‘state evolution’) that depends on the parameters defining the spatial coupling. Then, using a simple spatially coupled design and a judicious choice of functions for the AMP algorithm, we analyze the fixed points of the resulting state evolution and show that it achieves the asymptotic MMSE. Numerical results for phase retrieval and rectified linear regression show that spatially coupled designs can yield substantially lower MSE than i.i.d. Gaussian designs at finite dimensions when used with AMP algorithms.
{"title":"Bayes-Optimal Estimation in Generalized Linear Models via Spatial Coupling","authors":"Pablo Pascual Cobo;Kuan Hsieh;Ramji Venkataramanan","doi":"10.1109/TIT.2024.3455228","DOIUrl":"10.1109/TIT.2024.3455228","url":null,"abstract":"We consider the problem of signal estimation in a generalized linear model (GLM). GLMs include many canonical problems in statistical estimation, such as linear regression, phase retrieval, and 1-bit compressed sensing. Recent work has precisely characterized the asymptotic minimum mean-squared error (MMSE) for GLMs with i.i.d. Gaussian sensing matrices. However, in many models there is a significant gap between the MMSE and the performance of the best known feasible estimators. We address this issue by considering GLMs defined via spatially coupled sensing matrices. We propose an efficient approximate message passing (AMP) algorithm for estimation and prove that with a simple choice of spatially coupled design, the MSE of a carefully tuned AMP estimator approaches the asymptotic MMSE as the dimensions of the signal and the observation grow proportionally. To prove the result, we first rigorously characterize the asymptotic performance of AMP for a GLM with a generic spatially coupled design. This characterization is in terms of a deterministic recursion (‘state evolution’) that depends on the parameters defining the spatial coupling. Then, using a simple spatially coupled design and a judicious choice of functions for the AMP algorithm, we analyze the fixed points of the resulting state evolution and show that it achieves the asymptotic MMSE. Numerical results for phase retrieval and rectified linear regression show that spatially coupled designs can yield substantially lower MSE than i.i.d. Gaussian designs at finite dimensions when used with AMP algorithms.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"8343-8363"},"PeriodicalIF":2.2,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194907","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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