  In this paper, a stochastic geometry based analytical framework is proposed
for secure simultaneous transmitting and reflecting reconfigurable intelligent
surface (STAR-RIS) assisted non-orthogonal multiple access (NOMA)
transmissions, where legitimate users (LUs) and eavesdroppers are randomly
distributed. Both the time-switching protocol (TS) and energy splitting (ES)
protocol are considered for the STAR-RIS. To characterize system performance,
the channel statistics are first provided, and the Gamma approximation is
adopted for general cascaded $\kappa$-$\mu$ fading. Afterward, the closed-form
expressions for both the secrecy outage probability (SOP) and average secrecy
capacity (ASC) are derived. To obtain further insights, the asymptotic
performance for the secrecy diversity order and the secrecy slope are deduced.
The theoretical results show that 1) the secrecy diversity orders of the strong
LU and the weak LU depend on the path loss exponent and the distribution of the
received signal-to-noise ratio, respectively; 2) the secrecy slope of the ES
protocol achieves the value of one, higher than the slope of the TS protocol
which is the mode operation parameter of TS. The numerical results demonstrate
that: 1) there is an optimal STAR-RIS mode operation parameter to maximize the
secrecy performance; 2) the STAR-RIS-NOMA significantly outperforms the
STAR-RIS-orthogonal multiple access.
