1. A signature apparatus for generating a signature text by using a commitment, wherein the commitment is a hash value of a set including a value to be committed, data including a pair of elements of a cyclic group associated with a discrete logarithm problem which is used as a public key, and a discrete logarithm of an order of the pair which is used as a secret key, the signature apparatus comprising: a committed vector selecting unit configured to select a committed vector associated with a first commitment; a first commitment calculating unit configured to calculate the first commitment; a basis vector calculating unit configured to calculate a basis vector; a second commitment calculating unit configured to calculate a power residue and calculates a second commitment; a vector challenge calculating unit configured to calculate a vector challenge; a vector response calculating unit configured to calculate a vector response by using the first commitment, a set used for calculating the power residue, the vector challenge, and the basis vector; and a memory configured to store the committed vector, the first commitment, the basis vector, the second commitment, the vector challenge, and the vector response, wherein a signature text is generated based in part on the first commitment, the second commitment, and the vector response, and wherein a signature text output means is configured to read the signature text from the memory and output the signature text to a verifying apparatus, wherein the basis vector and the vector challenge are hash values, wherein the committed vector selecting unit is further configured to select a plurality of committed vectors, each having the same configuration as the committed vector, and wherein each component of the plurality of committed vectors and the secret key satisfy a relation equation, and wherein the first commitment is a hash value of data including components of the vector response, the public key which is data including a pair of elements of the cyclic group associated with the discrete logarithm problem, and the secret key which is a discrete logarithm of an order of the pair.