Regular expressions are widely used in various areas of computer science. However, due to the complex syntax and the use of a large number of meta-characters, regular expressions are quite error-prone when defined and used by developers. Testing is a practical and effective way to ensure the semantic correctness of regular expressions. The most common method is to generate a set of character strings according to the tested expression and check whether they comply with the intended language. Most of the existing test data generation focuses only on positive strings. However, empirical study shows that a majority of errors during actual development are manifested by the fact that the defined language is smaller than the intended one. In addition, such errors can only be detected by negative strings. This study investigates the generation of negative strings from regular expressions based on mutation. The study first obtains a set of mutants by injecting defects into the tested expression through mutation and then selects a negative character string in the complementary set of the language defined by the tested expression to reveal the error simulated by the corresponding mutant. In order to simulate complex defects and avoid the problem that the negative strings cannot be obtained due to the specialization of mutants, a second-order mutation mechanism is adopted. Meanwhile, optimization techniques such as redundant mutant elimination and mutation operator selection are used to reduce the mutants, so as to control the size of the finally generated test set. The experimental results show that the proposed algorithm can generate negative test strings with a moderate size and have strong error detection ability compared with the existing tools.