• Publications
  • Influence
Generating Hard Instances of Lattice Problems
  • M. Ajtai
  • Mathematics, Computer Science
  • Electron. Colloquium Comput. Complex.
  • 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 isExpand
  • 642
  • 88
Generating hard instances of lattice problems (extended abstract)
  • M. Ajtai
  • Computer Science
  • STOC '96
  • 1 July 1996
We give a random class of lattices in Zn whose elements can be generated together with a short vector in them so that, if there is a probabilistic polynomial time algorithm which finds a short vectorExpand
  • 745
  • 70
  • Open Access
A Note on Ramsey Numbers
Abstract Upper bounds are found for the Ramsey function. We prove R(3, x) cx 2 ln x and, for each k ⩾ 3, R(k, x) c k x k − 1 ( ln x) k − 2 asymptotically in x .
  • 304
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  • 413
  • 44
A public-key cryptosystem with worst-case/average-case equivalence
We present a probabilistic public key cryptosystem which is secure unless the worst case of the following lattice problem can be solved in polynomial time: “Find the shortest nonzero vector in an nExpand
  • 611
  • 42
  • Open Access
A sieve algorithm for the shortest lattice vector problem
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 thisExpand
  • 479
  • 37
  • Open Access
Crossing-Free Subgraphs
If m⩾4 then every planar drawing of a graph with n vertices and m edges contains more than m 3 /100 n 2 edge-crossings and fewer than 10 13n crossing-free subgraphs. The first result settles aExpand
  • 312
  • 27
On optimal matchings
Givenn random red points on the unit square, the transportation cost between them is tipically √n logn.
  • 175
  • 26
∑11-Formulae on finite structures
  • M. Ajtai
  • Mathematics, Computer Science
  • Ann. Pure Appl. Log.
  • 1 July 1983
  • 508
  • 25
An 0(n log n) sorting network
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 ofExpand
  • 570
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