On the arithmetic complexity of euler function

Abstract

It is shown that the problem of computing the Euler function is closely related to the problem of computing the permanent of a matrix as well as to the derandomization of the Identity Testing problem. Specifically, it is shown that (1) if computing the Euler function over a finite field is hard then computing permanent over the integers is also hard, and (2) if computing any factor of the Euler function over a field is hard then the Identity Testing problem over the field can be derandomized. © 2011 Springer-Verlag Berlin Heidelberg.