Non-orthogonal multiple access (NOMA) is a promising solution for secure transmission under massive access. However, in addition to the uncertain channel state information (CSI) of the eavesdroppers due to their passive nature, the CSI of the legitimate users may also be imperfect at the base station due to the limited feedback. Under both channel uncertainties, the optimal power allocation and transmission rate design for a secure NOMA scheme is currently not known due to the difficulty of handling the probabilistic constraints. This paper fills this gap by proposing novel transformation of the probabilistic constraints and variable decoupling so that the security guaranteed sum-rate maximization problem can be solved by alternatively executing branch-and-bound method and difference of convex programming. To scale the solution to a truly massive access scenario, a first-order algorithm with very low complexity is further proposed. Simulation results show that the proposed first-order algorithm achieves identical performance to the conventional method but saves at least two orders of magnitude in computation time. Moreover, the resultant transmission scheme significantly improves the security guaranteed sum-rate compared to the orthogonal multiple access transmission and NOMA ignoring CSI uncertainty.