D. Micciancio. The hardness of the closest vector problem with preprocessing. IEEE Transactions on Information Theory, 47(3):1212-1215, Mar. 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....Solving the subset sum problem amounts to find a 0, 1 solution of an inhomogeneous linear equation, which can be naturally viewed as a closest vector problem (see section 7.3. 2) A deterministic polynomial time reduction from the subset sum problem to the closest vector problem is given in [Mic01] In the case of low density knapsacks, one can derive from this reduction a provable method to solved the problem in polynomialtime with high probability [CJL 92] Basically all knapsack cryptosystems have been broken, either by specific (often lattice based) attacks or by the low density ....

D. Micciancio. The hardness of the closest vector problem with preprocessing, 2001.

....et al. 2] show that approximating NCP to within 2 log (1 ) for any 0 is hard under the assumption that NP QP . We also de ne NCPP as the preprocessing variant of NCP. Bruck and Naor [6] show that the NCPP problem is NP hard to solve exactly. This was later extended by Micciancio [13] to the CVPP. However, both results apply only to the exact version of the problems and as noted in [13] it is not clear how to extend them to hardness of approximation. The rst inapproximability result is due to Feige and Micciancio [10] There, it is shown that NCPP over any eld GF (q) is ....

....that NP QP . We also de ne NCPP as the preprocessing variant of NCP. Bruck and Naor [6] show that the NCPP problem is NP hard to solve exactly. This was later extended by Micciancio [13] to the CVPP. However, both results apply only to the exact version of the problems and as noted in [13], it is not clear how to extend them to hardness of approximation. The rst inapproximability result is due to Feige and Micciancio [10] There, it is shown that NCPP over any eld GF (q) is NP hard to approximate within constant factors less than 5=3 and that as a result CVPP is NP hard to ....

D. Micciancio. The hardness of the closest vector problem with preprocessing. IEEE Transactions on Information Theory, 47(3):1212-1215, 2001.

.... Kannan proved in [78, Section 7] that any algorithm approximating SVP to within a non decreasing function f(d) can be used to approximate CVP to within d 3=2 f(d) CVP was shown to be NPhard as early as in 1981 [49] for a much simpler one line proof using the knapsack problem, see [100]) Approximating CVP to within a quasi polynomial factor 2 log 1 Gamma is NP hard [7, 45] However, NP hardness results for SVP and CVP have limits. Goldreich and Goldwasser [58] showed that approximating SVP or CVP to within d= log d is not NP hard, unless the polynomial time hierarchy ....

....that any lattice vector whose distance to y is exactly n=4 is necessarily of the form (y 1 Gammax 1 ; yn Gammax n ) where s = i=1 x i a i and x i 2 f0; 1g. This gives a deterministic polynomial time reduction from the knapsack problem to CVP (this reduction appeared in [100] with a slightly different lattice) One can derive from this reduction a provable method to solve the knapsack problem in polynomial time with high probability when the knapsack density defined as d = n= max 1in log 2 a i is low (see [85, 51, 54] Indeed, if kx Gamma yk = n=4 is strictly ....

D. Micciancio. The hardness of the closest vector problem with preprocessing. IEEE Trans. Inform. Theory, 47(3):1212--1215, 2001.

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D. Micciancio. The hardness of the closest vector problem with preprocessing. IEEE Transactions on Information Theory, 47(3):1212-1215, Mar. 2001.

....minfkv wk : w 2 L(B)g. In the closest vector problem (CVP) one is given a basis B and a target vector v (usually not in the lattice) and must nd the lattice vector in L(B) closest to v. CVP was proved NP hard in [36] and it remains hard even if the lattice basis can be arbitrarily preprocessed [22], or one allows for approximate solutions with approximation factor 2 lg 1 n [3, 9] To date, the best polynomial time algorithm to approximate CVP achieves only a worst case approximation factor which is almost exponential in the dimension of the lattice [19, 4, 29] A closely related ....

D. Micciancio. The hardness of the closest vector problem with preprocessing. IEEE Transactions on Information Theory, 2001. To Appear.

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D. Micciancio. The hardness of the closest vector problem with preprocessing, 2001.

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