D. Micciancio. On the Hardness of the Shortest Vector Problem. Ph. D. Thesis, MIT, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....present all the main technical ideas of the proofs, and the remaining gaps can be filled by relatively easy, if tedious, arguments. To keep the article brief and focussed, our references are not exhaustive for a more complete bibliography, the reader is referred to the original papers and to [5, 12]. Finally, we invite the readers attention to role of randomness that is pervasive in all the results presented. To the best of our knowledge, randomness is only a recent phenomenon in the study of lattice problems. 2 Preliminaries Let Z, Z q , and R denote the integers, the integers modulo q, ....

....E contains f0; 1g n . We could embed our instance of subset sum in these coordinates and complete the reduction this already gives a non uniform reduction from subset sum to SVP. The next major step in the proof is to (probabilistically) make this uniform. Here, the version due to Micciancio [12] is much simpler, and is based on the following idea: let C also stand for the m Theta jCj matrix whose columns are the sequences in the family C; if P is a random n Theta m 0 1 matrix, each of whose entries is set to 1 independently with probability p (for a suitably chosen p) then P Delta C ....

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D. Micciancio. On the Hardness of the Shortest Vector Problem. Ph. D. Thesis, MIT, 1998.

...., b s be a set of linearly independent vectors in IR s . The set of vectors L = z : z = s X i=1 c i b i , c 1 , c s # ZZ is called an s dimensional full rank lattice. The set b 1 , b s is called a basis of L. It has been remarked in Section 2. 1 of [17] and then in Section 2.4 of [21] that the following statement holds which is somewhat stronger than that usually used in the literature. 4 Lemma 3 There exists a polynomial time algorithm which, for a given lattice L and a vector r = r 1 , r s ) # IR s , finds a lattice vector v = ....

D. Micciancio, On the hardness of the shortest vector problem, PhD Thesis, MIT, 1998.

....1 , b s be a set of linearly independent vectors in IR s . The set of vectors L = z : z = s X i=1 c i b i , c 1 , c s # ZZ is called an s dimensional full rank lattice. The set b 1 , b s is called a basis of L. It has been remarked in Section 2. 1 of [20] and then in Section 2.4 of [24] that the following statement holds which is somewhat stronger than that usually used in the literature. Lemma 5 There exists a polynomial time algorithm which, for given a lattice L and a vector r = r 1 , r s ) # IR s , finds a lattice vector v = v ....

D. Micciancio, On the hardness of the shortest vector problem, PhD Thesis, MIT, 1998.

..... b s be a set of linearly independent vectors in IR s . The set of vectors L = z : z = s X i=1 t i b i , t 1 , t s # ZZ is called an s dimensional full rank lattice. The set b 1 , b s is called the basis of L. It has been remarked in in Section 2. 1 of [8] and then in Section 2.4 of [10] that the following statement holds which is somewhat stronger than that usually used in the literature. Lemma 3.1. There exists a polynomial time algorithm which, for given a lattice L and a vector r = r 1 , r s ) # IR s , finds a lattice vector v = ....

D. Micciancio, On the hardness of the shortest vector problem, PhD Thesis, MIT, 1998.

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D. Micciancio, On the hardness of the shortest vector problem, PhD Thesis, MIT, 1998.

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