Based on the theory of quadratic residues

the algebra structure of RSA arithmetic is researched in this paper.This work calculates numbers of quadratic residues and non-residues in the group

Z

*

n

and investigates their relationship.

Z

*

n

is divided up by the group made up with all quadratic residues in

Z

*

n

and all cosets form a quotient group of order 4 which is a Klein group.Studyed the structure of strong RSA further

it shows that the element of order

(

n

)/2 exists and the group

Z

*

n

can be generated by three e

lements of quadratic non-residues.Let the facterization

n=p·q

the order of each element can be calculated

and the biggest order of all element is

lcm

(

p

-1

q

-1) in

Z

*

n

. It also shows how to find the element of the biggest order.So the algebra structure of RSA arithmetic is solved.