  In this work, the problem of transmitting an i.i.d Gaussian source over an
i.i.d Gaussian wiretap channel with an i.i.d Gaussian side information is
considered. The intended receiver is assumed to have a certain minimum SNR and
the eavesdropper is assumed to have a strictly lower SNR compared to the
intended receiver. The objective is minimizing the distortion of source
reconstruction at the intended receiver. In this work, it is shown that unlike
the Gaussian wiretap channel without side information, Shannon's source-channel
separation coding scheme is not optimum in the sense of achieving the minimum
distortion. Three hybrid digital-analog secure joint source channel coding
schemes are then proposed which achieve the minimum distortion. The first
coding scheme is based on Costa's dirty paper coding scheme and wiretap channel
coding scheme when the analog source is not explicitly quantized. The second
coding scheme is based on the superposition of the secure digital signal and
the hybrid digital-analog signal. It is shown that for the problem of
communicating a Gaussian source over a Gaussian wiretap channel with side
information, there exists an infinite family of optimum secure joint
source-channel coding scheme. In the third coding scheme, the quantized signal
and the analog error signal are explicitly superimposed. It is shown that this
scheme provides an infinite family of optimum secure joint source-channel
channel coding schemes with a variable number of binning. Finally, the proposed
secure hybrid digital-analog schemes are analyzed under the main channel SNR
mismatch. It is proven that the proposed schemes can give a graceful
degradation of distortion with SNR under SNR mismatch, i.e., when the actual
SNR is larger than the designed SNR.
