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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)

- 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)

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."

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)

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)

Let G be a (k + I)-graph (a hypergraph with each hyperedge of size k + 1) with n vertices and average degree t. Assume k Q t Q n. If G is uncrowded (contains no cycle of size 2, 3, dr 4) then there exists an independent set of size c,(n/t)(ln t)'lk. Let G be a graph with n vertices and average valence t. Turan's theorem implies a(G) > n/(t + 1). (See… (More)