Random mappings with restricted preimages

Authors:
Andrew MacFie;Daniel Panario
Affiliations:
School of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada;School of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada
Venue:
LATINCRYPT'12 Proceedings of the 2nd international conference on Cryptology and Information Security in Latin America
Year:
2012

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Abstract

In this paper we refer to finite endofunctions where each image has at most r preimages as r-mappings. Probabilistic analysis of these mappings provides heuristics for problems arising in cryptography which involve similar but more complicated classes of mappings. We give asymptotic probabilities and expectations related to the joint distribution of some pairs of mapping parameters for these mappings, extending work done previously for general, unrestricted mappings. We give an elementary derivation of the expected value of the maximum of tail length and cycle length of a random node in a random r-mapping, which is a useful component in the heuristic analysis of Pollard's rho algorithm. All distributions considered are uniform.