We pose and investigate the distributed secure source coding based on the common key cryptosystem. This cryptosystem includes the secrecy amplification problem for distributed encrypted sources with correlated keys using post-encryption-compression, which was posed investigated by Santoso and Oohama. In this paper we propose another new security criterion which is generally more strict compared to the commonly used security criterion which is based on the upper-bound of mutual information between the plaintext and the ciphertext. Under this criterion, we establish the necessary and sufficient condition for the secure transmission of correlated sources.