A cryptographic processing system comprising: a terminal, a cryptographic processing device and an authentication device, said terminal being adapted to receive second cryptographic information that represents an encrypted monomial obtained by encrypting a monomial generated by use of a random number using an homomorphic encryption; said cryptographic processing device comprising a storage configured to store therein first cryptographic information that represents an encrypted polynomial obtained by encrypting a first polynomial that corresponds to a first vector using said homomorphic encryption; a receiver configured to receive, from the terminal third cryptographic information that represents an encrypted result obtained by encrypting a result of a multiplication of a second polynomial that corresponds to a second vector by the monomial using the homomorphic encryption; a generator configured to generate an encrypted polynomial that includes a result of an operation of the first vector and the second vector by use of the first cryptographic information, the third cryptographic information, and information that represents the monomial; an output unit configured to output fourth cryptographic information that represents the encrypted polynomial that includes the result of the operation; and a transmitter configured to transmit the second cryptographic information to the terminal, wherein the generator is configured to generate (213, 509) a second encrypted polynomial obtained by encrypting the second polynomial by multiplying the encrypted result represented by the third cryptographic information and an inverse of the monomial, and to generate the encrypted polynomial that includes the result of the operation by multiplying the encrypted polynomial obtained by encrypting the first polynomial by the encrypted polynomial obtained by encrypting the second polynomial, and the output unit is configured to output the fourth cryptographic information to said authentication device; wherein said authentication device is configured to decrypt the fourth cryptographic information and to perform authentication processing on the basis of the result of the operation. a terminal, a cryptographic processing device and an authentication device, said terminal being adapted to receive second cryptographic information that represents an encrypted monomial obtained by encrypting a monomial generated by use of a random number using an homomorphic encryption; said cryptographic processing device comprising a storage configured to store therein first cryptographic information that represents an encrypted polynomial obtained by encrypting a first polynomial that corresponds to a first vector using said homomorphic encryption; a receiver configured to receive, from the terminal third cryptographic information that represents an encrypted result obtained by encrypting a result of a multiplication of a second polynomial that corresponds to a second vector by the monomial using the homomorphic encryption; a generator configured to generate an encrypted polynomial that includes a result of an operation of the first vector and the second vector by use of the first cryptographic information, the third cryptographic information, and information that represents the monomial; an output unit configured to output fourth cryptographic information that represents the encrypted polynomial that includes the result of the operation; and a transmitter configured to transmit the second cryptographic information to the terminal, wherein the generator is configured to generate (213, 509) a second encrypted polynomial obtained by encrypting the second polynomial by multiplying the encrypted result represented by the third cryptographic information and an inverse of the monomial, and to generate the encrypted polynomial that includes the result of the operation by multiplying the encrypted polynomial obtained by encrypting the first polynomial by the encrypted polynomial obtained by encrypting the second polynomial, and the output unit is configured to output the fourth cryptographic information to said authentication device; wherein said authentication device is configured to decrypt the fourth cryptographic information and to perform authentication processing on the basis of the result of the operation.