According to one embodiment, a representation converting unit converts a set of n elements (h0, h1, . . . , hn−1) (hi: a member of a finite field Fp^m, 0≦i≦n−1) that is a projective representation of a member g of an n-th degree algebraic torus Tn(Fp^m) (n: positive integer, p: prime number, m: positive integer) into a limited projected representation expressed by a set of n elements (h′0, h′1, . . . , h′n−1) (h′i: a member of the finite field Fp^m, 0≦i≦n−1) in which at least one element out of the n elements is a zero element 0 or an identity element 1. An arithmetic unit omits part of Fp^m operation that is arithmetic operation in the finite field Fp^m based on a fact that an element in the set of n elements (h′0, h′1, . . . , h′n−1) represented by the limited projective representation is a zero element “0” or an identity element “1” when performing Fp^mn operation that is arithmetic operation of a finite field Fp^mn in combination with the Fp^m operation.