Recently, Boneh et al. proposed an interesting algorithm for factoring integers, the so-called LFM (Lattice Factoring Method). It is based on the techniques of Coppersmith and Howgrave-Graham, namely, it cleverly empl...
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Recently, Boneh et al. proposed an interesting algorithm for factoring integers, the so-called LFM (Lattice Factoring Method). It is based on the techniques of Coppersmith and Howgrave-Graham, namely, it cleverly employs the lll-algorithm. The LFM is for integers of the form N = p(r)q, and is very effective for large r. That is, it runs in polynomial time in log N when r is on the order of logp. We note that for small r, e.g. N = pq, p(2)q, it is an exponential time algorithm in log N. In this paper, we propose a method for speeding up the LFM from a practical viewpoint. Also, theoretical considerations and experimental results are provided that show that the proposed algorithm offers shorter runing time than the original LFM.
The aim of this paper is a reduction algorithm for a basis b(1), b(2), b(3) of a 3-dimensional lattice in R-n for fixed n greater than or equal to 3. We give a definition of the reduced basis which is equivalent to th...
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ISBN:
(数字)9783540446705
ISBN:
(纸本)3540424881
The aim of this paper is a reduction algorithm for a basis b(1), b(2), b(3) of a 3-dimensional lattice in R-n for fixed n greater than or equal to 3. We give a definition of the reduced basis which is equivalent to that of the Minkowski reduced basis of a 3-dimensional lattice. We prove that for b(1), b(2), b(3) epsilon Z(n), n greater than or equal to 3 and \b(1)\, \b(2)\, \b(3)\ less than or equal to M, our algorithm takes O(log(2) M) binary operations, without using fast integer arithmetic, to reduce this basis and so to find the shortest vector in the lattice. The definition and the algorithm can be extended to any dimension. Elementary steps of our algorithm axe rather different from those of the lll-algorithm, which works in O(log(3) M) binary operations without using fast integer arithmetic.
We review public-key cryptosystems from lattice problems, which are inspired by Ajtai's remarkable result, and consider their security from the point of view of both theory and practice. We also survey recent resu...
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We review public-key cryptosystems from lattice problems, which are inspired by Ajtai's remarkable result, and consider their security from the point of view of both theory and practice. We also survey recent results on the power of the lattice reduction algorithm in cryptanalysis.
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