  This paper is concerned with several security notions for information
theoretically secure encryptions defined by the variational (statistical)
distance. To ensure the perfect secrecy (PS), the mutual information is often
used to evaluate the statistical independence between a message and a
cryptogram. On the other hand, in order to recognize the information
theoretically secure encryptions and computationally secure ones
comprehensively, it is necessary to reconsider the notion of PS in terms of the
variational distance. However, based on the variational distance, three kinds
of definitions for PS are naturally introduced, but their relations are not
known. In this paper, we clarify that one of three definitions for PS with the
variational distance, which is a straightforward extension of Shannon's perfect
secrecy, is stronger than the others, and the weaker two definitions of PS are
essentially equivalent to the statistical versions of indistinguishability and
semantic security.
