\emph{Information Dispersal Algorithms (IDAs)} have been widely applied to reliable and secure storage and transmission of data files in distributed systems. An IDA is a method that encodes a file $F$ of size $L=|F|$ into $n$ unrecognizable pieces $F_1$, $F_2$, ..., $F_n$, each of size $L/m$ ($m<n$), so that the original file $F$ can be reconstructed from any $m$ pieces. The core of an IDA is the adopted non-systematic $m$-of-$n$ erasure code. This paper makes a systematic study on the \emph{confidentiality} of an IDA and its connection with the adopted erasure code. Two levels of confidentiality are defined: \emph{weak confidentiality} (in the case where some parts of the original file $F$ can be reconstructed explicitly from fewer than $m$ pieces) and \emph{strong confidentiality} (in the case where nothing of the original file $F$ can be reconstructed explicitly from fewer than $m$ pieces). For an IDA that adopts an arbitrary non-systematic erasure code, its confidentiality may fall into weak confidentiality. To achieve strong confidentiality, this paper explores a sufficient and feasible condition on the adopted erasure code. Then, this paper shows that Rabin's IDA has strong confidentiality. At the same time, this paper presents an effective way to construct an IDA with strong confidentiality from an arbitrary $m$-of-$(m+n)$ erasure code. Then, as an example, this paper constructs an IDA with strong confidentiality from a Reed-Solomon code, the computation complexity of which is comparable to or sometimes even lower than that of Rabin's IDA.