This investigation proposes two methods for embedding backdoors in the RSA modulus N=pq rather than in the public exponent e. This strategy not only permits manufacturers to embed backdoors in an RSA system, but also allows users to choose any desired public exponent, such as e=2¹⁶+1, to ensure efficient encryption. This work utilizes lattice attack and exhaustive attack to embed backdoors in two proposed methods, called RSA_SBLT and RSA_SBES, respectively. Both approaches involve straightforward steps, making their running time roughly the same as that of normal RSA key-generation time, implying that no one can detect the backdoor by observing time imparity.