| Summary: | 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.
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