A computing apparatus outputs τ1 and τ2 corresponding to a ciphertext x, a capability providing apparatus uses τ1 to correctly compute f(τ1) with a probability greater than a certain probability and sets the result of the computation as z1, uses τ2 to correctly compute f(τ2) with a probability greater than a certain probability and sets the result of the computation as z2, the computing apparatus generates a computation result u=f(x)bx1 from z1, generates a computation result v=f(x)ax2 from z2, and outputs ub′va′ if the computation results u and v satisfy a particular relation, where G and H are groups, f(x) is a function for obtaining an element of the group G for xεH, X1 and X2 are random variables having values in the group G, x1 is a realization of the random variable X1, and x2 is a realization of the random variable X2.