1. An encryption processing method in a key-insulated cryptosystem which includes terminal devices each for carrying out encryption processing, and external devices each for updating a terminal secret key stored in each of the terminal devices and for storing secret information that is a secret key selected from a predetermined number of pairs of a public key and a secret key, wherein the key-insulated cryptosystem includes at least first and second external devices, the first and second external devices are associated with a number of times the terminal secret key has been updated, and a different piece of secret information is stored in each of the first and second external devices, the encryption processing method comprising: selecting one of the first and second external devices based on the number of times the terminal secret key has been updated, causing the selected one of the first and second external devices to generate key-updating information used for updating the terminal secret key based on the number of times the terminal secret key has been updated, and the stored secret information, causing the terminal device to update the terminal secret key to a post-first-update terminal secret key by using first key-updating information generated by the first external device, causing the terminal device to update the terminal secret key to a post-second-update terminal secret key by using second key-updating information generated by the second terminal device, at the timing of an update immediately once the post-first-update terminal secret key is generated, causing the terminal device to encrypt information, by using a first public key corresponding to the post-first-update terminal secret key and by using a second public key corresponding to the post-second-update terminal secret key, and causing another terminal device different from the terminal device to decrypt the information, by using the post-first-update terminal secret key and the post-second-update terminal secret key.