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It is known that the equivalence of interleavers for turbo codes using quadratic permutation polynomials (QPPs) over integer rings can be exactly determined by the so-called quadratic null polynomials (QNPs) over integer rings. For generating QNPs or higher order null polynomials (NPs), some theoretical results have been obtained in previous literature. In this letter, it is proved that the coefficients of previously obtained QNPs are not only sufficient but also necessary for generating any QNPs. Based on the necessary and sufficient conditions for generating QNPs and QPPs, the enumeration of QPPs excluding their equivalence is presented. The obtained results are helpful to investigate the algebraic structure of QPP interleavers as well as to avoid the equivalence in the design of QPP interleavers.