| Running time predictions for factoring algorithms (2008) | |||||||||||||||
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| Partiellement soutenu par une bourse de la Conseil de recherches en sciences naturelles et en génie du Canada. 3 Supported in part by NSF Grant DMS-01-03635. In 1994, Carl Pomerance proposed the following problem: Select integers a1, a2,..., aJ at random from the interval [1, x], stopping when some (non-empty) subsequence, {ai: i ∈ I} where I ⊆ {1, 2,..., J}, has a square product (that is ∏ i∈I ai ∈ Z2). What can we say about the possible stopping times, J? A 1985 algorithm of Schroeppel can be used to show that this process stops after selecting (1 + ɛ)J0(x) integers aj with probability 1 − o(1) (where the function J0(x) is given explicitly in (1) below). Schroeppel’s algorithm actually finds the square product, and this has subsequently been adopted, with relatively minor modifications, by all factorers. In 1994 Pomerance showed that, with probability 1−o(1), the | |||||||||||||||
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