Download Complexity of Lattice Problems: A Cryptographic Perspective by Daniele Micciancio PDF

By Daniele Micciancio

The ebook offers a self-contained assessment of the state-of-the-art within the complexity of lattice difficulties, with specific emphasis on difficulties which are relating to the development of cryptographic services. particular subject matters coated are the most powerful identified inapproximability outcome for the shortest vector challenge; the relatives among this and different computational lattice difficulties; an exposition of the way cryptographic services will be equipped and turn out safe in line with worst-case hardness assumptions approximately lattice difficulties; and a learn of the boundaries of non-approximability of lattice difficulties. a few heritage in complexity conception, yet no previous wisdom approximately lattices, is thought.

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Extra info for Complexity of Lattice Problems: A Cryptographic Perspective

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12) is strictly less than 1. It follows that for all sufficiently large n, (2. 12) is less than 1 , and 1 - e ( - ( 1 /n) C ) � (3/4) ( 1 - (3/4) 1 / ( n- 1 ) ) . 12) is 0. Let x = 1/(n - 1 ) and substitute n = 1 + 1 /x in (2. 13) � (1 - (it) We compute the limit of ( 2 . 13) for x -+ 0. Both the numerator and the denominator tend to 0, so we can use L'Hopital's rule and obtain __L c - 1 e- ( x/ ( l+ x )) C � + x )) c l (x ( e ( / 1 + r \l x r l + x ) =r =0 X� i ( 1 - (it) X� i ( �) X ln(4/3) • 0 This proves that with polynomially many iterations, the LLL al­ gorithm approximates the shortest vector in a lattice within a factor ( 2 /VJ)n .

1 di . ( Notice that this is the same integer defined in {2. 1 1) . ) But we know from the analysis of the number of iterations that log d is initially bounded by a polynomial in the input size, and it can only decrease during the execution of the algorithm. So, the denominators of all rational numbers occurring during the computation have polynomial size. It remains to show that also the magnitude of the numbers is polyno­ mial. We already know that J-li,j are at most 1/2 in absolute value. We now bound the length of the vectors.

When used in the BKZ algorithm, (Aj tai et al. , 2001) allows to reduce the SVP approximation factor from 2°(n(lnInn)2 /Inn) to 2°(n InInn/ Inn), although the output of the algorithm is only guaranteed to be short with high probability. 44 COMPLEXITY OF LATTICE PROBLEMS The nearest plane algorithm of Section 3 is one of two CVP approxi­ mation algorithms presented and analyzed in (Babai, 1 986) . The other algorithm, called the "rounding off" algorithm, simply expresses the tar­ get vector t = Bx in terms of the LLL reduced lattice basis B, rounds each coordinate of X to the closest integer Yi = fxiJ, and outputs lattice vector By.

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