A cryptographic system comprising: a key generation apparatus to generate, as a secret key for an input parameter Y, a secret key including a matrix e as a key element, wherein a product of the matrix e and a matrix A Y determined by the input parameter Y forms a matrix u j for a randomly selected value j in a set including a plurality of values, the matrix u j being among a plurality of matrices u obtained from public parameters PP; an encryption apparatus to generate a ciphertext including cipher elements for respective values included in the set; and a decryption apparatus to decrypt the ciphertext with using the secret key and the cipher element for the value j among cipher elements included in the ciphertext, wherein the matrix A Y is a matrix A v in which a predicate vector v being the input parameter Y is set, wherein the encryption apparatus generates a ciphertext that includes a cipher element in which an attribute vector w is set, in addition to the cipher elements for the respective values, and wherein the key generation apparatus generates the matrix e indicated in formula 1, as the key element, wherein the encryption apparatus generates c 0 indicated in formula 2, c i,γ indicated in formula 3, and c' i indicated in formula 4, as the cipher elements, and wherein the decryption apparatus decrypts the ciphertext by conducting calculation indicated in formula 5 A v ⋅ e = u j mod q, A v: = A ‖ C v ∈ Z q n × 2 m, A ∈ Z q n × m, C v: = ∑ i = 1 L ∑ γ = 0 k v i, γ A i, γ ∈ Z q n × m, v i = ∑ γ = 0 k v i, γ ⋅ 2 γ, v: = v 1,..., v L, where each ν i,γ is 0 or 1, where n is an integer not less than 1, m is an integer not less than n, q is an integer larger than 2, and L is an integer not less than 1 c 0: = A T s + X, s ← U Z q n, X ← R φ ̅ α m, where φ is a probability distribution over Z q:= Z / q Z c i, γ: = A i, γ + 2 γ w i B T s + R i, γ T X, w: = w 1,..., w L, B ← U Z q n × m, R i, γ T ∈ 0,1 m × m c i ′: = u i T s + x + M ⋅ " q / 2 " ∈ Z q for i ∈ N, x ← R φ ̅ α, M ∈ 0,1, where " q /2" is an integer that is the closest to q /2 c v: = ∑ i = 1 L ∑ γ = 0 k v i, γ c i, γ, c: = c 0 ‖ c v, z: = c j ′ − e T c mod q where each α is a positive real-number Gaussian parameter, in the above formulae. a key generation apparatus to generate, as a secret key for an input parameter Y, a secret key including a matrix e as a key element, wherein a product of the matrix e and a matrix A Y determined by the input parameter Y forms a matrix u j for a randomly selected value j in a set including a plurality of values, the matrix u j being among a plurality of matrices u obtained from public parameters PP; an encryption apparatus to generate a ciphertext including cipher elements for respective values included in the set; and a decryption apparatus to decrypt the ciphertext with using the secret key and the cipher element for the value j among cipher elements included in the ciphertext, wherein the matrix A Y is a matrix A v in which a predicate vector v being the input parameter Y is set, wherein the encryption apparatus generates a ciphertext that includes a cipher element in which an attribute vector w is set, in addition to the cipher elements for the respective values, and wherein the key generation apparatus generates the matrix e indicated in formula 1, as the key element, wherein the encryption apparatus generates c 0 indicated in formula 2, c i,γ indicated in formula 3, and c' i indicated in formula 4, as the cipher elements, and wherein the decryption apparatus decrypts the ciphertext by conducting calculation indicated in formula 5 where each ν i,γ is 0 or 1, where n is an integer not less than 1, m is an integer not less than n, q is an integer larger than 2, and L is an integer not less than 1 n is an integer not less than 1, m is an integer not less than n, q is an integer larger than 2, and L is an integer not less than 1