This paper investigates the secrecy energy-efficiency of a multi-user downlink non-orthogonal multiple access (NOMA) transmission in the presence of a passive eavesdropper. The secrecy energy-efficiency, capturing the trade-off between the achievable secrecy sum-rate and the power consumption, is here formulated as a bi-criterion optimization problem that is shown to be convex. As such, the characterization of the Pareto-boundary boils down to maximizing the scalarized trade-off between the secrecy sum-rate and the power consumption, which is solved in closed-form. Finally, this obtained closed-form is exploited to maximize the ratio between the secrecy sum-rate and the power consumption. Our numerical results highlight the relative secrecy energy-efficiency gain between NOMA and OMA that can reach up to 550%.