Interesting Esoterica

How long does it take to catch a wild kangaroo?

Article by Ravi Montenegro and Prasad Tetali
  • Published in 2008
  • Added on
We develop probabilistic tools for upper and lower bounding the expected time until two independent random walks on $\ZZ$ intersect each other. This leads to the first sharp analysis of a non-trivial Birthday attack, proving that Pollard's Kangaroo method solves the discrete logarithm problem $g^x=h$ on a cyclic group in expected time $(2+o(1))\sqrt{b-a}$ for an average $x\in_{uar}[a,b]$. Our methods also resolve a conjecture of Pollard's, by showing that the same bound holds when step sizes are generalized from powers of 2 to powers of any fixed $n$.

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BibTeX entry

@article{Howlongdoesittaketocatchawildkangaroo,
	title = {How long does it take to catch a wild kangaroo?},
	abstract = {We develop probabilistic tools for upper and lower bounding the expected time
until two independent random walks on {\$}\ZZ{\$} intersect each other. This leads to
the first sharp analysis of a non-trivial Birthday attack, proving that
Pollard's Kangaroo method solves the discrete logarithm problem {\$}g^x=h{\$} on a
cyclic group in expected time {\$}(2+o(1))\sqrt{\{}b-a{\}}{\$} for an average
{\$}x\in{\_}{\{}uar{\}}[a,b]{\$}. Our methods also resolve a conjecture of Pollard's, by
showing that the same bound holds when step sizes are generalized from powers
of 2 to powers of any fixed {\$}n{\$}.},
	url = {http://arxiv.org/abs/0812.0789v2 http://arxiv.org/pdf/0812.0789v2},
	year = 2008,
	author = {Ravi Montenegro and Prasad Tetali},
	comment = {},
	urldate = {2018-05-12},
	archivePrefix = {arXiv},
	eprint = {0812.0789},
	primaryClass = {math.PR},
	collections = {animals,easily-explained,probability-and-statistics}
}