On Quadratic Inverses for Quadratic Permutation Polynomials over Integer   Rings
  An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementation. Sun and Takeshita have recently shown that the class of quadratic permutation polynomials over integer rings provides excellent performance for turbo codes. In this correspondence, a necessary and sufficient condition is proven for the existence of a quadratic inverse polynomial for a quadratic permutation polynomial over an integer ring. Further, a simple construction is given for the quadratic inverse. All but one of the quadratic interleavers proposed earlier by Sun and Takeshita are found to admit a quadratic inverse, although none were explicitly designed to do so. An explanation is argued for the observation that restriction to a quadratic inverse polynomial does not narrow the pool of good quadratic interleavers for turbo codes. 
Jonghoon Ryu and Oscar Y. Takeshita