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- 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 is… Expand

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 vector… Expand

A Note on Ramsey Numbers

- M. Ajtai, J. Komlós, E. Szemerédi
- Mathematics, Computer Science
- J. Comb. Theory, Ser. A
- 1 November 1980

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 .

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 n… Expand

A sieve algorithm for the shortest lattice vector problem

- M. Ajtai, Ravi Kumar, D. Sivakumar
- Mathematics, Computer Science
- STOC '01
- 6 July 2001

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… Expand

Crossing-Free Subgraphs

- M. Ajtai, V. Chvátal, M. Newborn, E. Szemerédi
- Mathematics
- 1982

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 a… Expand

On optimal matchings

- M. Ajtai, J. Komlós, G. Tusnády
- Mathematics, Computer Science
- Comb.
- 1 December 1984

Givenn random red points on the unit square, the transportation cost between them is tipically √n logn.

∑11-Formulae on finite structures

- M. Ajtai
- Mathematics, Computer Science
- Ann. Pure Appl. Log.
- 1 July 1983

An 0(n log n) sorting network

- M. Ajtai, J. Komlós, E. Szemerédi
- Computer Science
- STOC '83
- 1 December 1983

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… Expand