The main aim of class discriminative universal adversarial perturbations (CD-UAPs) is that the adversary can flexibly control the targeted class and influence remaining classes limitedly. CD-UAPs generated by the existing attack strategies suffer from a high fooling ratio of non-targeted source classes under non-targeted and targeted attacks, and face the increasing risk of discovery. In this paper, we propose a training framework for generating enhanced covertness CD-UAPs. It trains the targeted source class set and the non-targeted source classes set alternately to update the perturbation and introduces logit pairing to mitigate the influence of perturbation on the non-targeted source classes set. Further, we extend CD-UAPs on the targeted (one-targeted) attack to the multi-targeted attack, which perturbs a targeted source class to multiple targeted sink classes that seriously threaten the current scenario. It can not only provide the adversary with freedom of precise attack but reduce the risk of being detected. This attack poses a strong threat to security-sensitive applications. Extensive experiments on the CIFAR-10, CIFAR-100 and ImageNet datasets show our method can generate more deceptive perturbations and enhance the covertness of CD-UAPs. For example, our method improves the absolute fooling ratio gaps of ResNet-20 and VGG-16 by 9.46% and 6.94% compared with the baseline method, respectively. We achieve the multi-targeted attack with a high fooling ratio on the GTSRB dataset. The average absolute target fooling ratio gaps of ResNet-20 and VGG-16 are 81.89% and 76.33%, respectively.