Annals of Applied Probability

Collision times in multicolor urn models and sequential graph coloring with applications to discrete logarithms

Bhaswar B. Bhattacharya

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Consider an urn model where at each step one of $q$ colors is sampled according to some probability distribution and a ball of that color is placed in an urn. The distribution of assigning balls to urns may depend on the color of the ball. Collisions occur when a ball is placed in an urn which already contains a ball of different color. Equivalently, this can be viewed as sequentially coloring a complete $q$-partite graph wherein a collision corresponds to the appearance of a monochromatic edge. Using a Poisson embedding technique, the limiting distribution of the first collision time is determined and the possible limits are explicitly described. Joint distribution of successive collision times and multi-fold collision times are also derived. The results can be used to obtain the limiting distributions of running times in various birthday problem based algorithms for solving the discrete logarithm problem, generalizing previous results which only consider expected running times. Asymptotic distributions of the time of appearance of a monochromatic edge are also obtained for other graphs.

Article information

Ann. Appl. Probab., Volume 26, Number 6 (2016), 3286-3318.

Received: July 2015
Revised: January 2016
First available in Project Euclid: 15 December 2016

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Zentralblatt MATH identifier

Primary: 05C15: Coloring of graphs and hypergraphs 60F05: Central limit and other weak theorems
Secondary: 94A62: Authentication and secret sharing [See also 81P94] 60G55: Point processes

Discrete logarithm graph coloring limit theorems Poisson embedding


Bhattacharya, Bhaswar B. Collision times in multicolor urn models and sequential graph coloring with applications to discrete logarithms. Ann. Appl. Probab. 26 (2016), no. 6, 3286--3318. doi:10.1214/16-AAP1176.

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