SummaryFog computing is composed of various computers with weak performance instead of servers with strong performance. As history has shown, there has not been a general pairing‐free certificateless signature scheme that is mainly designed with modular exponentiation and modular multiplication that can possess resistance to Type I and Type II adversaries. The lightweight certificateless signature algorithm with low requirements for computing and storage capabilities, which can be practicably implemented in fog computing, needs to be studied. Therefore, a new hard mathematic problem is firstly defined in this paper, which is called variant of RSA problem. Then, a new general pairing‐free certificateless signature scheme is proposed based on the variant of RSA problem and the discrete logarithm problem. Fortunately, the proposed scheme is the first RSA‐based certificateless signature scheme that can possess resistance to Type I and Type II adversaries. A formal security proof is provided to demonstrate that, under adaptively chosen message attacks, the scheme is provably secure against Type I and Type II adversaries in the random oracle model. When compared with other known pairing‐free certificateless signature schemes of the same type, the computation cost of our scheme is slightly higher; however, a higher security level can be achieved.