R. Kannan. Algorithmic geometry of numbers. Annual Reviews in Computer Science, 2:231--267, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....factor rational polynomials in polynomial time (back then, a famous problem) from which the name LLL comes. Further refinements of the LLL algorithm were later proposed, notably by Schnorr [121, 122] Those algorithms have proved invaluable in many areas of mathematics and computer science (see [91, 78, 132, 64, 36, 84]) In particular, their relevance to cryptology was immediately understood, and they were used to break schemes based on the knapsack problem (see [119, 29] which were early alternatives to the RSA cryptosystem [120] The success of reduction algorithms at breaking various cryptographic schemes ....

....We will not discuss Ajtai s worst case average case equivalence [3, 33] which refers to special versions of SVP and SBP (see [30, 31, 14] such as SVP when the lattice gap 2 = 1 is at least polynomial in the dimension. 2. 4 Algorithmic results The main algorithmic results are surveyed in [91, 78, 132, 64, 36, 84, 30, 109]. No polynomial time algorithm is known for approximating either SVP, CVP or SBP to within a polynomial factor in the dimension d. In fact, the existence of such algorithms is an important open problem. The best polynomial time algorithms achieve only slightly subexponential factors, and are based ....

R. Kannan. Algorithmic geometry of numbers. Annual review of computer science, 2:231--267, 1987.

....factor rational polynomials in polynomial time (back then, a famous problem) from which the name LLL comes. Further refinements of the LLL algorithm were later proposed, notably by Schnorr [101, 102] Those algorithms have proved invaluable in many areas of mathematics and computer science (see [75, 64, 109, 52, 30, 69]) In particular, their relevance to The technique is however polynomial time for fixed dimension, which was enough in [74] cryptology was immediately understood, and they were used to break schemes based on the knapsack problem (see [99, 23] which were early alternatives to the RSA ....

....problem. CVP seems to be a more difficult problem. Goldreich et al. 50] recently noticed that CVP cannot be easier than SVP: given an oracle that approximates CVP to within a factor f(d) one can approximate SVP in polynomial time to within the same factor f(d) Reciprocally, Kannan proved in [64] 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 NP hard as early as in 1981 [40] for a simplified proof, see [65] Approximating CVP to within a quasi polynomial factor 2 log ....

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R. Kannan. Algorithmic geometry of numbers. Annual review of computer science, 2:231--267, 1987.

....CVP when the approximation factor is polynomial (or super polynomial) in the dimension, or the norm is not an p one. We recall that only when the approximation factor is almost exponential (2 O(n(lg lg n) 2 = lg n) the two problems are known to be solvable in polynomial time 2 (cf. [11,4,13,9]) The rst non empirical evidence that SVP is not harder than CVP (in the same dimension) was recently given by Henk [8] who showed that solving SVP (in the exact sense) is reducible to solving CVP (also in the exact sense) Moreover, the result in [8] holds for a wide variety of norms (not ....

R. Kannan, Algorithmic Geometry of numbers, Annual Reviews in Computer Science 2 (1987), pp. 231-267.

....factor rational polynomials in polynomial time (back then, a famous problem) from which the name LLL comes. Further refinements of the LLL algorithm were later proposed, notably by Schnorr [101, 102] Those algorithms have proved invaluable in many areas of mathematics and computer science (see [75, 64, 109, 52, 30, 69]) In particular, their relevance to 1 The technique is however polynomial time for fixed dimension, which was enough in [74] 2 cryptology was immediately understood, and they were used to break schemes based on the knapsack problem (see [99, 23] which were early alternatives to the RSA ....

....problem. CVP seems to be a more difficult problem. Goldreich et al. 50] recently noticed that CVP cannot be easier than SVP: given an oracle that approximates CVP to within a factor f(d) one can approximate SVP in polynomial time to within the same factor f(d) Reciprocally, Kannan proved in [64] 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) 2 . CVP was shown to be NP hard as early as in 1981 [40] for a simplified proof, see [65] Approximating CVP to within a quasi polynomial factor 2 log ....

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R. Kannan. Algorithmic geometry of numbers. Annual review of computer science, 2:231--267, 1987.

....except that polynomial time (approximation) algorithms were harder to find and stronger hardness results were more easily established. Babai [8] modified the LLL reduction algorithm to approximate in polynomial time CVP within a factor 2 n . The approximation factor was improved to 2 ffln in [69, 45, 71]. Kannan [46] gave a polynomial time algorithm to solve CVP exactly in any fixed number of dimensions. The dependency of the running time on the dimension is again 2 n ln n . Finding a polynomial time algorithm to approximate CVP within a polynomial factor is a major open problem in the area. ....

.... Kannan showed that approximating CVP within a factor p n is polynomial time Turing reducible to solving SVP exactly [46] and approximating SVP within a factor n 3=2 f(n) 2 is polynomial time Turing reducible to approximating SVP within a factor f(n) for any non decreasing function f(n) [45]. More will be said about reducing CVP to SVP in the next chapter, where we prove that SVP is NP hard to approximate by reduction from a modification of CVP. As an aside, 46] shows also that the search and decisional versions of SVP are polynomial time Turing equivalent. 23 2.2 Promise Problems ....

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R. Kannan. Algorithmic geometry of numbers. Annual Reviews in Computer Science, 2:231--267, 1987.

....The latter work also shows that if CVP could be approximated within any factor greater than 2 log 1 Gammaffl n , then NP e P. On the other hand, Babai showed that CVP can be approximated within factor 2 n by a modification of the LLL lattice reduction algorithm [8] and improvements by [45, 34] yield for every ffl 0 approximation within factor 2 ffln . The problem of verifying the approximate optimality of a solution to the CVP problem has also been considered. Given a point c in the lattice, its distance to t clearly provides an upper bound on the minimum distance of t to the ....

....a lattice in R n , v 2 R n is a vector, d 2 R and dist(v; L(B) g(n) Delta d. For any g 1, the promise problem GapCVP g is in NP (i.e. in the extension of NP to promise problems) The NP witness for (B; v; d) being a yes instance is merely a vector u 2 L(B) satisfying dist(v; u) d. By [40, 45, 34], GapCVP 2 ffln is decidable in polynomial time, for every ffl 0. No 1 An equivalent formulation used below refers to the minimum distance between a pair of distinct lattice points. 2 A promise problem is a pair, Pi yes ; Pi no ) of non intersecting subsets of f0; 1g . The subset Pi ....

R. Kannan. Algorithmic Geometry of Numbers. Annual Reviews in Computer Science, Vol. 2, pages 231--267, 1987. 19

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R. Kannan. Algorithmic geometry of numbers. Annual Reviews in Computer Science, 2:231--267, 1987.

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