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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)

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 dimensional lattice L where the shortest vector v is unique in the sense that any other vector whose length is at most n c kvk is parallel to v."

- Miklós Ajtai
- STOC
- 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 in a random lattice with a probability of at least ~ then there is also a probabilistic polynomial time algorithm which solves the following three lattice… (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)

I,aht)ral.t)ry I'or Compul,er Science. and I)t'lmrl.nnml. 1. Introduction I{t~txml, ly there has been much interest ill tht, comput:Ltional power of circuits o1' bounded dt;pth. In particular Furst, Saxe and Sipser [FSS], and indcpentlently Ajtai [Aj] in a different form, have shown that no polynomial size circuits of bounded depth can compute the parity… (More)