  The information-theoretic secure exact-repair regenerating codes for
distributed storage systems (DSSs) with parameters $(n,k=d,d,\ell)$ are studied
in this paper. We consider distributed storage systems with $n$ nodes, in which
the original data can be recovered from any subset of $k=d$ nodes, and the
content of any node can be retrieved from those of any $d$ helper nodes.
Moreover, we consider two secrecy constraints, namely, Type-I, where the
message remains secure against an eavesdropper with access to the content of
any subset of up to $\ell$ nodes, and Type-II, in which the message remains
secure against an eavesdropper who can observe the incoming repair data from
all possible nodes to a fixed but unknown subset of up to $\ell$ compromised
nodes. Two classes of secure determinant codes are proposed for Type-I and
Type-II secrecy constraints. Each proposed code can be designed for a range of
per-node storage capacity and repair bandwidth for any system parameters. They
lead to two achievable secrecy trade-offs, for Type-I and Type-II security.
