Miklós Ajtai

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  • Miklós Ajtai
  • Electronic Colloquium on Computational Complexity
  • 1996
We give a random class of lattices in Z n so that, if there is a probabilistic polynomial time algorithm which nds a short vector in a random lattice with a probability of at least 1 2 then there is also a probabilistic polynomial time algorithm which solves the following three lattice problems in every lattice in Z n with a probability exponentially close(More)
The purpose of this paper is to describe a sorting network of size 0(n log n) and depth 0(log n). A natural way of sorting is through consecutive halvings: determine the upper and lower halves of the set, proceed similarly within the halves, and so on. Unfortunately, while one can halve a set using only 0(n) comparisons, this cannot be done in less than(More)
We present a randomized 2^{<italic>O(n)</italic>} time algorithm to compute a shortest non-zero vector in an <italic>n</italic>-dimensional rational lattice. The best known time upper bound for this problem was 2^{<italic>O(n</italic>\log <italic>n</italic>)} first given by Kannan [7] in 1983. We obtain several consequences of this algorithm for related(More)
The Pigeonhole Principle for n is the statement that there is no one-to-one function between a set of size n and a set of size n 1. This statement can be formulated as an unlimited fan-in constant depth polynomial size Boolean formula PHPn in n ( n 1 ) variables. We may think that the truth-value of the variable xi, j will be true iff the function maps the(More)