Let [tex]x^2_1+x^2_2=1[/tex] be an unit circle upon a finite field [tex]Z_{p}[/tex] where p is a prime. Is there any algorithm (other than the brute force algorithm) which can give all the possible solutions [tex](x_1,x_2)\in Z_{p}\times Z_{p}[/tex] as well as the total number of such solutions? If exists, what is the complexity of it?(adsbygoogle = window.adsbygoogle || []).push({});

More generally, is there an answer of the same question, when, instead of a circle we consider a n-sphere [tex]x^2_1+x^2_2+\cdots+x^2_n=1[/tex]? What will happen if, instead of [tex]Z_{p}[/tex] we work with [tex]Z_{q}[/tex], where [tex]q=p^r[/tex]?

I am not a number theorist. So I do not know whether any thing exists in literature. Please help.

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# Points on an unit circle over finite field

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