By Mary Gray
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Extra resources for A Radical Approach to Algebra (Addison-Wesley Series in Mathematics)
Springer, Heidelberg (2001) 10. : Segment LLL-Reduction with Floating Point Orthogonalization. H. ) CaLC 2001. LNCS, vol. 2146, pp. 81–96. Springer, Heidelberg (2001) 11. : Factoring Polynomials with Rational Coeﬃcients. Math. Ann. 261, 515–534 (1982) 12. : Cryptanalysis of NTRU (preprint) (1999) 13. : Low-Dimensional Lattice Basis Reduction Revisited. A. ) Algorithmic Number Theory. LNCS, vol. 3076, pp. 338–357. Springer, Heidelberg (2004) 14. : Floating-Point LLL Revisited. F. ) EUROCRYPT 2005.
B2i,t(q2i ) xe2i,t(q2i ) , where e2i−1,1 = d(q2i−1 ) and e2i,1 = d(q2i ) and the exponent of each term in q2i−1 (x) is greater than the exponent of each term in q2i (x). If for all indices i = 1, 2, . . G. W. S. Vigklas for any permutation of the positive coeﬃcients c2i−1,j , j = 1, 2, . . , t(q2i−1 ). Otherwise, for each of the indices i for which we have t(q2i−1 ) < t(q2i ), we break up one of the coeﬃcients of q2i−1 (x) into t(q2i ) − t(q2i−1 ) + 1 parts, so that now t(q2i ) = t(q2i−1 ) and apply the same formula (3) given above.
4. Reduction times for fpLLL and xLiDIA using diﬀerent η at dimension 85 for M1 , dimension 95 for M2 , and dimension 85 for M3 , the reduction time of fpLLL even exceeds that of NTL. This behavior of fpLLL is due to the overhead caused by updating two matrices (Gram matrix and lattice basis) for each transformation in the reduction process. The newly-introduced concept of buﬀered transformations as part of xLiDIA prevents this kind of behavior. 5 as in the original LLL algorithm and the xLiDIA implementation used for Figures 1 – 3.
A Radical Approach to Algebra (Addison-Wesley Series in Mathematics) by Mary Gray