Leveled Homomorphic Encryption Based on Finite Field Isomorphism Problem Over Matrix Algebra

Finite fields of order p
^m
are isomorphic. The elements of these fields can be denoted by a polynomial with coefficients modulo p or by matrices with entries chosen modulo p. The length of an element of field can be associated with the norm of that element. In general, a non-trivial isomorphic map between any two fields does not preserve the length of the elements. The entries of the images of short elements distribute randomly and uniformly over modulo p. This key feature yields new kind of problems, called finite field isomorphism problems. In this paper, we studied the finite field isomorphism problem and introduced a leveled homomorphic encryption scheme based on the finite field isomorphism problem over the matrix field.