Thread: Show that a^2 -1 = 0 mod p has only 2 solutions

I have an exercise for a course in Cryptography. One of the questions asks us to show that the equation a^2 -1 = 0 mod p has only 2 solutions.

we are told to consider the group Z sub p whose elements are {1,2,.....,p-1}.

I am having a hard time even finding a starting point for this problem. Can anyone give me some advice?

Originally Posted by restin84

I have an exercise for a course in Cryptography. One of the questions asks us to show that the equation a^2 -1 = 0 mod p has only 2 solutions.

we are told to consider the group Z sub p whose elements are {1,2,.....,p-1}.

I am having a hard time even finding a starting point for this problem. Can anyone give me some advice?

Originally Posted by restin84

I have an exercise for a course in Cryptography. One of the questions asks us to show that the equation a^2 -1 = 0 mod p has only 2 solutions.

we are told to consider the group Z sub p whose elements are {1,2,.....,p-1}.

I am having a hard time even finding a starting point for this problem. Can anyone give me some advice?

No matter which is p>2, a=1 and a=p-1 always satisfy the congruence equation . In general if we have a congruence equation of the form...

(1)

... then the sum of molteplicities of the roots can't be greater than n...

Kind regards