Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution
This paper studies the secrecy capacity of an <i>n</i>-dimensional Gaussian wiretap channel under a peak power constraint. This work determines the largest peak power constraint <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"...
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MDPI AG
2023-04-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/5/741 |
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| author | Antonino Favano Luca Barletta Alex Dytso |
| author_facet | Antonino Favano Luca Barletta Alex Dytso |
| author_sort | Antonino Favano |
| collection | DOAJ |
| description | This paper studies the secrecy capacity of an <i>n</i>-dimensional Gaussian wiretap channel under a peak power constraint. This work determines the largest peak power constraint <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi mathvariant="sans-serif">R</mi><mo>¯</mo></mover><mi>n</mi></msub></semantics></math></inline-formula>, such that an input distribution uniformly distributed on a single sphere is optimal; this regime is termed the low-amplitude regime. The asymptotic value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi mathvariant="sans-serif">R</mi><mo>¯</mo></mover><mi>n</mi></msub></semantics></math></inline-formula> as <i>n</i> goes to infinity is completely characterized as a function of noise variance at both receivers. Moreover, the secrecy capacity is also characterized in a form amenable to computation. Several numerical examples are provided, such as the example of the secrecy-capacity-achieving distribution beyond the low-amplitude regime. Furthermore, for the scalar case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>=</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, we show that the secrecy-capacity-achieving input distribution is discrete with finitely many points at most at the order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><msup><mi mathvariant="sans-serif">R</mi><mn>2</mn></msup><msubsup><mi>σ</mi><mn>1</mn><mn>2</mn></msubsup></mfrac></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>σ</mi><mn>1</mn><mn>2</mn></msubsup></semantics></math></inline-formula> is the variance of the Gaussian noise over the legitimate channel. |
| first_indexed | 2024-03-11T03:45:42Z |
| format | Article |
| id | doaj.art-61d2a376a3c74e8fa4c6550d4a044ff0 |
| institution | Directory Open Access Journal |
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| language | English |
| last_indexed | 2024-03-11T03:45:42Z |
| publishDate | 2023-04-01 |
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| spelling | doaj.art-61d2a376a3c74e8fa4c6550d4a044ff02023-11-18T01:15:44ZengMDPI AGEntropy1099-43002023-04-0125574110.3390/e25050741Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input DistributionAntonino Favano0Luca Barletta1Alex Dytso2Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, 20133 Milano, ItalyDipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, 20133 Milano, ItalyQualcomm, Bridgewater, NJ 08807, USAThis paper studies the secrecy capacity of an <i>n</i>-dimensional Gaussian wiretap channel under a peak power constraint. This work determines the largest peak power constraint <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi mathvariant="sans-serif">R</mi><mo>¯</mo></mover><mi>n</mi></msub></semantics></math></inline-formula>, such that an input distribution uniformly distributed on a single sphere is optimal; this regime is termed the low-amplitude regime. The asymptotic value of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mover accent="true"><mi mathvariant="sans-serif">R</mi><mo>¯</mo></mover><mi>n</mi></msub></semantics></math></inline-formula> as <i>n</i> goes to infinity is completely characterized as a function of noise variance at both receivers. Moreover, the secrecy capacity is also characterized in a form amenable to computation. Several numerical examples are provided, such as the example of the secrecy-capacity-achieving distribution beyond the low-amplitude regime. Furthermore, for the scalar case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>=</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>, we show that the secrecy-capacity-achieving input distribution is discrete with finitely many points at most at the order of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><msup><mi mathvariant="sans-serif">R</mi><mn>2</mn></msup><msubsup><mi>σ</mi><mn>1</mn><mn>2</mn></msubsup></mfrac></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>σ</mi><mn>1</mn><mn>2</mn></msubsup></semantics></math></inline-formula> is the variance of the Gaussian noise over the legitimate channel.https://www.mdpi.com/1099-4300/25/5/741wiretap channelMIMOamplitude constraints |
| spellingShingle | Antonino Favano Luca Barletta Alex Dytso Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution Entropy wiretap channel MIMO amplitude constraints |
| title | Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution |
| title_full | Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution |
| title_fullStr | Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution |
| title_full_unstemmed | Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution |
| title_short | Amplitude Constrained Vector Gaussian Wiretap Channel: Properties of the Secrecy-Capacity-Achieving Input Distribution |
| title_sort | amplitude constrained vector gaussian wiretap channel properties of the secrecy capacity achieving input distribution |
| topic | wiretap channel MIMO amplitude constraints |
| url | https://www.mdpi.com/1099-4300/25/5/741 |
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