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Some New Identities on Kloosterman Sums

In this correspondence, we prove that the distributions of the two-dimensional Kloosterman sums are completely determined by the distributions of the (one-dimensional) Kloosterman sums, and, by a famous result of Lauchad and Wolfman, we can give all the values of the two-dimensional Kloosterman sums. Furthermore, using the identities on one-dimensional Kloosterman sums found by Tor Helleseth and Victor Zinoviev, Hollmann and Xiang, etc., we present some new identities on Kloosterman sums in this note. Meanwhile, we also provide a new approach to producing new Kloosterman sums' identities