The Shortest Vector in a Lattice is Hard to Approximate to Within Some Constant
نویسنده
چکیده
We show the shortest vector problem in the l2 norm is NP-hard (for randomized reductions) to approximate within any constant factor less than p2. We also give a deterministic reduction under a reasonable number theoretic conjecture. Analogous results hold in any lp norm (p 1). In proving our NP-hardness result, we give an alternative construction satisfying Ajtai’s probabilistic variant of Sauer’s lemma, that greatly simplifies Ajtai’s original proof.
منابع مشابه
On the Inapproximability of the Shortest Vector in a Lattice within some constant factor (preliminary version)
We show that computing the approximate length of the shortest vector in a lattice within a factor c is NP-hard for randomized reductions for any constant c < p 2. email: [email protected]. Partially supported by DARPA contract DABT63-96-C-0018.
متن کاملThe Shortest Vector in a Lattice is
We show that computing the approximate length of the shortest vector in a lattice within a factor c is NP-hard for randomized reductions for any constant c < p 2. We also give a deterministic reduction based on a number theoretic conjecture.
متن کاملApproximating-CVP to Within Almost-Polynomial Factors is NP-Hard
This paper shows the closest vector in a lattice to be NPhard to approximate to within any factor up to 2(logn)1 where = (log logn) c for any constant c < 12 . Introduction A lattice L = L(v1; ::; vn), for vectors v1; ::; vn 2 Rn is the set of all integer linear combinations of v1; ::; vn, that is, L = fP aivi j ai 2 Zg. Given a lattice L and an arbitrary vector y, the Closest Vector Problem (C...
متن کاملOn the hardness of the shortest vector problem
An n-dimensional lattice is the set of all integral linear combinations of n linearly independent vectors in ' tm. One of the most studied algorithmic problems on lattices is the shortest vector problem (SVP): given a lattice, find the shortest non-zero vector in it. We prove that the shortest vector problem is NP-hard (for randomized reductions) to approximate within some constant factor great...
متن کاملThe Hardness of Approximate Optima in Lattices , Codes , and Systems of Linear
We prove the following about the Nearest Lattice Vector Problem (in any`p norm), the Nearest Codeword Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NP-hard. 2. If for some > 0 there exists a polynomial-time algorithm that approximates the optimum within a factor of 2 lo...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- SIAM J. Comput.
دوره 30 شماره
صفحات -
تاریخ انتشار 1998