  In this paper, we provide expressions for the secrecy outage probability
(SOP) for suboptimal and optimal opportunistic scheduling schemes in a
reconfigurable intelligent surface (RIS) aided system with multiple
eavesdroppers in approximate closed form. A suboptimal scheduling (SS) scheme
is analyzed, which is used when the channel state information (CSI) of the
eavesdropping links is unavailable, and the optimal scheduling (OS) scheme is
also analyzed, which is used when the global CSI is available. For each scheme,
we provide a simplified expression for the SOP in the high signal-to-noise
ratio (SNR) regime to demonstrate its behavior as a function of the key system
parameters. At high SNR, the SOP saturates to a constant level which decreases
exponentially with the number of RIS elements in the SS scheme and with the
product of the number of RIS elements and the number of users in the OS scheme.
We compare the performance of the opportunistic user scheduling schemes with
that of a non-orthogonal multiple access (NOMA) based scheduling scheme which
chooses a pair of users in each time slot for scheduling and we show that the
opportunistic schemes outperform the NOMA-based scheme. We also derive a
closed-form expression for the SOP of a decode-and-forward (DF) relay-aided
scheduling scheme in order to compare it with that of the RIS-aided system. It
is found that the RIS-aided system outperforms the relay-aided systems when the
number of RIS elements is sufficiently large. An increased number of RIS
elements is required to outperform the relay-aided system at higher operating
frequencies.
