PURPOSE: To authenticate a message with security at a high speed by encrypting a residue calculated through the use of an exchanged encryption key, transmitting the residue, and certifying the received message, only when the value obtained by decoding and calculating the received residue is a determined value. CONSTITUTION: A copy of a message M11 is loaded to a virtual buffer 12. Then an arithmetic unit 13 using a random number polynomial p(x) with respect to an expanded message M' in a buffer 12 is used for residue arithmetic operation. Then an encryption device 14 using an encryption key exchanged between a sender and a recipient encrypts the residue. Then the residue is linked with a message M in a virtual buffer 15, and the resulting message is sent to the recipient. The recipient gives the received message and the encrypted residue to the virtual buffer, where the encrypted residue is decoded by using the the exchanged encrypted key. Then the decoded residue is linked with the received message. Residue arithmetic operation is applied to the polynomial, and when the result is zero, the message is accepted.