This paper studies the secrecy capacity of ann-dimensional Gaussian wiretap channel under a peak power constraint. This work determines the largest peak power constraintR¯n, such that an input distribution uniformly distributed on a single sphere is optimal; this regime is termed the low-amplitude regime. The asymptotic value ofR¯nasngoes 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(n=1), we show that the secrecy-capacity-achieving input distribution is discrete with finitely many points at most at the order ofR2σ12, whereσ12is the variance of the Gaussian noise over the legitimate channel.