{"title":"关于包络核问题随机实例的解数","authors":"Carlo Sanna","doi":"10.1016/j.jco.2024.101898","DOIUrl":null,"url":null,"abstract":"<div><div>The <em>Permuted Kernel Problem</em> (PKP) is a problem in linear algebra that was first introduced by Shamir in 1989. Roughly speaking, given an <span><math><mi>ℓ</mi><mo>×</mo><mi>m</mi></math></span> matrix <strong><em>A</em></strong> and an <span><math><mi>m</mi><mo>×</mo><mn>1</mn></math></span> vector <strong><em>b</em></strong> over a finite field of <em>q</em> elements <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, the PKP asks to find an <span><math><mi>m</mi><mo>×</mo><mi>m</mi></math></span> permutation matrix <strong><em>π</em></strong> such that <span><math><mi>π</mi><mi>b</mi></math></span> belongs to the kernel of <strong><em>A</em></strong>. In recent years, several post-quantum digital signature schemes whose security can be provably reduced to the hardness of solving random instances of the PKP have been proposed. In this regard, it is important to know the expected number of solutions to a random instance of the PKP in terms of the parameters <span><math><mi>q</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>m</mi></math></span>. Previous works have heuristically estimated the expected number of solutions to be <span><math><mi>m</mi><mo>!</mo><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>ℓ</mi></mrow></msup></math></span>.</div><div>We provide, and rigorously prove, exact formulas for the expected number of solutions to a random instance of the PKP and the related <em>Inhomogeneous Permuted Kernel Problem</em> (IPKP), considering two natural ways of generating random instances.</div></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":"86 ","pages":"Article 101898"},"PeriodicalIF":1.8000,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0885064X2400075X/pdfft?md5=939873f4b51043507214927d47f2bb37&pid=1-s2.0-S0885064X2400075X-main.pdf","citationCount":"0","resultStr":"{\"title\":\"On the number of solutions to a random instance of the permuted kernel problem\",\"authors\":\"Carlo Sanna\",\"doi\":\"10.1016/j.jco.2024.101898\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>The <em>Permuted Kernel Problem</em> (PKP) is a problem in linear algebra that was first introduced by Shamir in 1989. Roughly speaking, given an <span><math><mi>ℓ</mi><mo>×</mo><mi>m</mi></math></span> matrix <strong><em>A</em></strong> and an <span><math><mi>m</mi><mo>×</mo><mn>1</mn></math></span> vector <strong><em>b</em></strong> over a finite field of <em>q</em> elements <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, the PKP asks to find an <span><math><mi>m</mi><mo>×</mo><mi>m</mi></math></span> permutation matrix <strong><em>π</em></strong> such that <span><math><mi>π</mi><mi>b</mi></math></span> belongs to the kernel of <strong><em>A</em></strong>. In recent years, several post-quantum digital signature schemes whose security can be provably reduced to the hardness of solving random instances of the PKP have been proposed. In this regard, it is important to know the expected number of solutions to a random instance of the PKP in terms of the parameters <span><math><mi>q</mi><mo>,</mo><mi>ℓ</mi><mo>,</mo><mi>m</mi></math></span>. Previous works have heuristically estimated the expected number of solutions to be <span><math><mi>m</mi><mo>!</mo><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>ℓ</mi></mrow></msup></math></span>.</div><div>We provide, and rigorously prove, exact formulas for the expected number of solutions to a random instance of the PKP and the related <em>Inhomogeneous Permuted Kernel Problem</em> (IPKP), considering two natural ways of generating random instances.</div></div>\",\"PeriodicalId\":50227,\"journal\":{\"name\":\"Journal of Complexity\",\"volume\":\"86 \",\"pages\":\"Article 101898\"},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2024-09-24\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0885064X2400075X/pdfft?md5=939873f4b51043507214927d47f2bb37&pid=1-s2.0-S0885064X2400075X-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Complexity\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0885064X2400075X\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Complexity","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0885064X2400075X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the number of solutions to a random instance of the permuted kernel problem
The Permuted Kernel Problem (PKP) is a problem in linear algebra that was first introduced by Shamir in 1989. Roughly speaking, given an matrix A and an vector b over a finite field of q elements , the PKP asks to find an permutation matrix π such that belongs to the kernel of A. In recent years, several post-quantum digital signature schemes whose security can be provably reduced to the hardness of solving random instances of the PKP have been proposed. In this regard, it is important to know the expected number of solutions to a random instance of the PKP in terms of the parameters . Previous works have heuristically estimated the expected number of solutions to be .
We provide, and rigorously prove, exact formulas for the expected number of solutions to a random instance of the PKP and the related Inhomogeneous Permuted Kernel Problem (IPKP), considering two natural ways of generating random instances.
期刊介绍:
The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited.
Areas Include:
• Approximation theory
• Biomedical computing
• Compressed computing and sensing
• Computational finance
• Computational number theory
• Computational stochastics
• Control theory
• Cryptography
• Design of experiments
• Differential equations
• Discrete problems
• Distributed and parallel computation
• High and infinite-dimensional problems
• Information-based complexity
• Inverse and ill-posed problems
• Machine learning
• Markov chain Monte Carlo
• Monte Carlo and quasi-Monte Carlo
• Multivariate integration and approximation
• Noisy data
• Nonlinear and algebraic equations
• Numerical analysis
• Operator equations
• Optimization
• Quantum computing
• Scientific computation
• Tractability of multivariate problems
• Vision and image understanding.