This work examines the properties of code sequences for the degraded scalar Gaussian broadcast channel with confidential messages (BCC) in terms of the behavior of the mutual information and minimum mean-square error (MMSE) functions for all signal-to-noise ratios (SNRs). More specifically, the work focuses on both completely secure code sequences, meaning that the transmitted message to the stronger receiver, i.e., with higher SNR, is completely secure in the equivocation sense, and code sequences with maximum equivocation. In these two cases an alternative converse proof is provided which also depicts the exact behavior of the relevant MMSE functions for all SNRs, for 'good', capacity achieving, code sequences. This means that the amount of disturbance on unintended receivers that is inflicted by 'good' code sequences is fully depicted. Moreover, the work also considers the effect that MMSE constraints, which limit the amount of disturbance on some unintended receiver, have on the capacity region of a completely secure code sequence. For maximum rate-pairs complying with the MMSE constraint, the behavior of the relevant MMSE functions for all SNRs is fully depicted.