Currently, the Internet of Things (IoT) provides individuals with real-time data processing and efficient data transmission services, relying on extensive edge infrastructures. However, those infrastructures may disclose sensitive information of consumers without authorization, which makes data access control to be widely researched. Ciphertext-policy attribute-based encryption (CP-ABE) is regarded as an effective cryptography tool for providing users with a fine-grained access policy. In prior ABE schemes, the attribute universe is only managed by a single trusted central authority (CA), which leads to a reduction in security and efficiency. In addition, all attributes are considered equally important in the access policy. Consequently, the access policy cannot be expressed flexibly. In this paper, we propose two schemes with a new form of encryption named multi-authority criteria-based encryption (CE) scheme. In this context, the schemes express each criterion as a polynomial and have a weight on it. Unlike ABE schemes, the decryption will succeed if and only if a user satisfies the access policy and the weight exceeds the threshold. The proposed schemes are proved to be secure under the decisional bilinear Diffie–Hellman exponent assumption (q-BDHE) in the standard model. Finally, we provide an implementation of our works, and the simulation results indicate that our schemes are highly efficient.