By Daniele Micciancio

ISBN-10: 1461352932

ISBN-13: 9781461352938

ISBN-10: 1461508975

ISBN-13: 9781461508977

Lattices are geometric gadgets that may be pictorially defined because the set of intersection issues of an enormous, average n-dimensional grid. De spite their obvious simplicity, lattices disguise a wealthy combinatorial struc ture, which has attracted the eye of significant mathematicians over the past centuries. no longer unusually, lattices have discovered a number of ap plications in arithmetic and desktop technological know-how, starting from quantity idea and Diophantine approximation, to combinatorial optimization and cryptography. The learn of lattices, particularly from a computational standpoint, was once marked by means of significant breakthroughs: the improvement of the LLL lattice aid set of rules by way of Lenstra, Lenstra and Lovasz within the early 80's, and Ajtai's discovery of a connection among the worst-case and average-case hardness of convinced lattice difficulties within the past due 90's. The LLL set of rules, regardless of the rather negative caliber of the answer it provides within the worst case, allowed to plot polynomial time ideas to many classical difficulties in laptop technology. those contain, fixing integer courses in a set variety of variables, factoring polynomials over the rationals, breaking knapsack dependent cryptosystems, and discovering strategies to many different Diophantine and cryptanalysis problems.

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**Additional info for Complexity of Lattice Problems: A Cryptographic Perspective**

**Example text**

This completes the proof that the vectors of a reduced basis are as short as possible. 2 Gauss' algorithm In this subsection we describe an algorithm to find a reduced basis for any 2-dimensionallattice. 3, works by computing a sequence of bases satisfying the following property. 2 A basis [a, b] is well ordered if lIall ::; lIa - bll < IIbli. c([a, b]). This is easily accomplished by a simple case analysis. {See part of the 28 COMPLEXITY OF LATTICE PROBLEMS Input: two linearly independent vectors a and b.

21 Basics deduces that (B, r) is a YES instance. On the other hand, assume one has a decision oracle A that solves GAPSVP,.. ) Let u E Z be an upper bound to 'x(B)2 (for example, let u be the squared length of any of the basis vectors). Notice that A(B, JU) always returns YES, while A(B, 0) always returns NO. Using binary search find an integer r E {O, ... ,u} such that A(B, Jr) = YES and A(B, vr=T) = NO. Then, 'xl (B) must lie in the interval [Jr, 'Y . Jr). A similar argument holds for the closest vector problem.

This proves that d decreases at least by a factor 0 at each iteration. Let do be the integer associated to the input matrix, and let dk be the integer associated to B after k iterations. By induction on k, dk ~ 8kdo. Since dk is a positive integer, 8kdo ~ Ok ~ 1 and for any 0 < 1 it must be Indo k ~ In{1/8)" Since do is computable in polynomial time from B, In do is clearly polynomial in the input size. If 8 is set to any fixed constant less than 1, then the (In{ 1/8)) -1 factor increases the number of iterations only by a constant factor.

### Complexity of Lattice Problems: A Cryptographic Perspective by Daniele Micciancio

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