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An algorithm for anonymous sharing of private data among N parties is developed. This technique is used iteratively to assign these nodes ID numbers ranging from 1 to N. This assignment is anonymous in that the identities received are unknown to the other members of the group. Resistance to collusion among other members is verified in an information theoretic sense when private communication channels are used. This assignment of serial numbers allows more complex data to be shared and has applications to other problems in privacy preserving data mining, collision avoidance in communications and distributed database access. The required computations are distributed without using a trusted central authority. Existing and new algorithms for assigning anonymous IDs are examined with respect to trade-offs between communication and computational requirements. The new algorithms are built on top of a secure sum data mining operation using Newton's identities and Sturm's theorem. An algorithm for distributed solution of certain polynomials over finite fields enhances the scalability of the algorithms. Markov chain representations are used to find statistics on the number of iterations required, and computer algebra gives closed form results for the completion rates.