Parallel Construction of Irreducible Polynomials

Let arithmetic pseudo-NCk denote the problems that can be solved by log space uniform arithmetic circuits over the finite prime field Fp of depth O(log(n + p)) and size (n + p)O(1). We show that the problem of constructing an irreducible polynomial of specified degree over Fp belongs to pseudo-NC2.5. We prove that the problem of constructing an irreducible polynomial of specified degree over Fp whose roots are guaranteed to form a normal basis for the corresponding field extension pseudo-NC2-reduces to the problem of factor refinement. We show that factor refinement of polynomials is in arithmetic NC3. Our algorithm works over any field and compared to other known algorithms it does not assume the ability to take p’th roots when the field has characteristic p. CR Categories: F.2.1. This research was supported by the ESPRIT II Basic Research Actions Program of the EC under contract No. 3075 (project ALCOM). Department of Computer Science, Aarhus University, Ny Munkegade, 8000 Aarhus C, Denmark. [email protected]