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Integrated sensing and communication (ISAC) is expected to be offered as a fundamental service in the upcoming sixth-generation (6G) communications standard. However, due to the exposure of information-bearing signals to the sensing targets, ISAC poses unique security challenges. In recent years, intelligent reflecting surfaces (IRSs) have emerged as a novel hardware technology capable of enhancing the physical layer security of wireless communication systems. Therefore, in this paper, we consider the problem of transmit and reflective beamforming design in a secure IRS-enabled ISAC system to maximize the beampattern gain at the target. The formulated non-convex optimization problem is challenging to solve due to the intricate coupling between the design variables. Moreover, alternating optimization (AO) based methods are inefficient in finding a solution in such scenarios, and convergence to a stationary point is not theoretically guaranteed. Therefore, we propose a novel successive convex approximation (SCA)-based second-order cone programming (SOCP) scheme in which all of the design variables are updated simultaneously in each iteration. The proposed SCA-based method significantly outperforms a penalty-based benchmark scheme previously proposed in this context. Moreover, we also present a detailed complexity analysis of the proposed scheme, and show that despite having slightly higher per-iteration complexity than the benchmark approach the average problem-solving time of the proposed method is notably lower than that of the benchmark scheme.