Our goal is to determine 
. Let us assume we have a prime number p and that
p-1 can be factored in the following way:
then we can say that
Thus if one can find 
for all i, one could use the Chinese Remainder Theorem to
combine them and obtain 
. In order to do this consider expanding
as follows
Then it is possible to show that
In order to show this, we have to see that
Multiplying out Eq (6) one obtains
Applying 
Eq(7) (one can apply 
to Eq(7) since it represents the
exponent of Eq(5) which is all done in 
)
Therefore Eq(5) becomes
Let 
and 
. Then
Using Shank's algorithm we find 
given 
, 
,
and p (notice that there are only 
possible values for 
). Now let
. Then
Considering only the exponents 
Then,
and we can use Shank's algorithm to find 
given 
,
, and p. We
now repeat the same procedure to obtain 
as follows.
Doing p-1 modular arithmetic, one finds that
Then if one lets
We then apply Shank's algorithm to find 
. This process is
continued to obtain all 
coefficients in Eq(4). From these
coefficients, we can find 
for all 
and finally combine them using the Chinese Remainder Theorem to
obtain 
. [4]