Finding Solutions for x12+x22=1 on Finite Fields Zp using Prime Number Algorithm

[khotsofalang]

i am sorry guys, the last time i posted this problem it was completely different but this time if we
Let x₁²+x₂²=1 be a unit circle upon a finite field Zp where p is prime. Is there any algorithm which can give all the possible solutions (x₁,x₂) an element of Zp*Zp as well as the total number of such solutions? If exists, what is the complexity of it?

 

[CRGreathouse]

You could check all possibilities. That takes something like O(p^2 log^2 p).

Now you just need a *good* way to solve it.

 

[khotsofalang]

all right, but what i actually need is that good way of solving it

 

[khotsofalang]

i need a solution to such an equation for stregthening my extended essay,anibody with a gud way of solving it?

 

[dodo]

If I understood correctly, you are looking for two quadratic residues that add up to 1. It may be easier for primes congruent to 1 modulo 4, because quadratic residues for those primes are 'symmetric': r is a quadratic residue iif p-r is. In this case you just look for contiguous quadratic residues on the lower half, from 2 to (p-1)/2: if r and r+1 are quadratic residues, then p-r also is, and (p-r)+(r+1) add to 1. My 2 cents.