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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: a = MatrixSpace(ZZ,3)(2); a [2 0 0] [0 2 0] [0 0 2] sage: a = matrix(ZZ,1,3, [1,2,-3]); a [ 1 2 -3] sage: a = MatrixSpace(ZZ,2,4)(2); a Traceback (most recent call last): ... TypeError: nonzero scalar matrix must be square BKZ(delta=None, fp=’rr’, block_size=10, prune=0, use_givens=False) Block Korkin-Zolotarev reduction. INPUT: •fp - ‘fp’ - double precision: NTL’s FP or fpLLL’s double •’qd’ - quad doubles: NTL’s QP •’qd1’ - quad doubles: uses quad_float precision to compute Gram-Schmidt, but uses double precision in the search phase of the block reduction algorithm. This seems adequate for most purposes, and is faster than ‘qd’, which uses quad_float precision uniformly throughout. •’xd’ - extended exponent: NTL’s XD •’rr’ - arbitrary precision (default) •block_size - specifies the size of the blocks in the reduction. High values yield shorter vectors, but the running time increases exponentially with block_size. block_size should be between 2 and the number of rows of self (default: 10) •prune - The optional parameter prune can be set to any positive number to invoke the Volume Heuristic from [Schnorr and Horner, Eurocrypt ‘95]. This can significantly reduce the running time, and hence allow much bigger block size, but the quality of the reduction is of course not as good in general. Higher values of prune mean better quality, and slower running time. When prune == 0, pruning is disabled. Recommended usage: for block_size = 30, set 10 = prune = 15. •use_givens - use Given’s orthogonalization. This is a bit slower, but generally much more stable, and is really the preferred orthogonalization strategy. For a nice description of this, see Chapter 5 of [G. Golub and C. van Loan, Matrix Computations, 3rd edition, Johns Hopkins Univ. Press, 1996]. EXAMPLE: sage: A = Matrix(ZZ,3,3,range(1,10)) sage: A.BKZ() [ 0 0 0] [ 2 1 0] [-1 1 3] sage: A = Matrix(ZZ,3,3,range(1,10)) sage: A.BKZ(use_givens=True) [ 0 0 0] [ 2 1 0] [-1 1 3] sage: A = Matrix(ZZ,3,3,range(1,10)) sage: A.BKZ(fp="fp") [ 0 0 0] [ 2 1 0] [-1 1 3] 314 Chapter 17. Dense matrices over the integer ring
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 LLL(delta=None, eta=None, algorithm=’fpLLL:wrapper’, fp=None, prec=0, early_red=False, use_givens=False) Returns LLL reduced or approximated LLL reduced lattice R for this matrix interpreted as a lattice. A lattice (b 1 , b 2 , ..., b d ) is (δ, η) -LLL-reduced if the two following conditions hold: •For any i > j, we have |mu i,j | and b∗ i is the i-th vector of the Gram-Schmidt orthogonalisation of (b 1 , b 2 , ..., b d ). The default reduction parameters are δ = 3/4 and eta = 0.501. The parameters δ and η must satisfy: 0.25 < δ
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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