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The following is from wikipedia:

The lifting formulas below, however, are often as good as an explicit evaluation. If $gcd(a,p) = 1$ one also has the important transformation: $$S(a,a;p) = \sum_{m=0}^{p-1} \left(\frac{m^2-4a^2}{p}\right) e^{\frac{2\pi i m}{p}},$$ where $\left(\tfrac{\ell}{m}\right)$ denotes the Jacobi symbol.

I am looking for the reference of this formula.

I am also looking for similar formulas for other Kloosterman sums ( higher dimensions).

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    $\begingroup$ This one's easy enough to reconstruct an argument without a reference: For how many $x$ is $m = ax + ax^{-1}$? This gives a quadratic equation with discriminant $m^2-4a^2$, etc. $\endgroup$ – Noam D. Elkies Mar 11 '17 at 2:33
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Regarding your first request on a reference for the formula:

This can be found in several places, but here is one where its online with an easy access. The paper is by Albert L. Whiteman, A note on Kloosterman sums, Bull. Amer. Math. Soc. 51 (1945), 373--377. It has derivations as well.

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