Intelligent reflecting surface (IRS) and non-orthogonal multiple access (NOMA) are both powerful means to improve the spectrum efficiency and can assist the cognitive radio network (CRN) in achieving more effective spectrum allocation. However, spectrum sharing in CRNs and the utilization of the scarce bandwidth in NOMA make it vulnerable to security threats. The secure beamforming design in IRS-assisted CRNs with NOMA is complicated and non-convex. Traditional semidefinite relaxation along with Gaussian randomization (SDR-GR) algorithm can solve the non-convex problem, however a high computational complexity. This paper proposes an alternating direction method of multipliers (ADMM)-based sum secrecy rate maximization (A-SSRM) scheme to achieve secure transmission with low computation complexity in IRS-assisted uplink CRNs with NOMA. Specifically, we derive an appropriate transformation of the original problem based on the ADMM framework and decompose it into multiple sub-problems, which are carefully designed to have closed-form solutions. Furthermore, an eavesdropper zero forcing (EZF)-based sub-optimal scheme is presented. Simulation results reveal that our proposed A-SSRM scheme outperforms other benchmarks in terms of sum secrecy rate and reduces the average running time by 6 times compared with the traditional SDR-GR algorithm. This is attributed to our effective decoupling and decomposing of the original problem, which enables each sub-problem to obtain a simple closed-form solution through derivation.