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

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

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

Givenn random red points on the unit square, the transportation cost between them is tipically √n logn, where logn is the number of red points in the ellipsoidal plane.Expand

Several consequences of this algorithm for related problems on lattices and codes are obtained, including an improvement for polynomial time approximations to the shortest vector problem.Expand

The pigeonhole principle can be formulated as an unlimited-fan-in constant depth polynomial-size Boolean formula PHP/sub n/ in n(n-1) variables, and it cannot be proved in the propositional calculus if the size of the proof isPolynomial in n and each formula is constant-depth, polynometric size and contains only the variables of PHP/ sub n/.Expand